Interpolatory Nonlinear Model Order Reduction

and its Application in Circuit Simulation

by

Seyed-Ali Nouri

A thesis submitted to the Faculty of Graduate and Postdoctoral Affairs

in partial fulfillment of the requirements for the degree of

Master of Applied Science

in

Electrical and Computer Engineering

Carleton University

Ottawa, Ontario, Canada

© 2021

Seyed-Ali Nouri

Abstract

This thesis presents a new approach to construct parametrized reduced-order models

for nonlinear circuits. The reduced model is obtained such that it matches the vari-

ations in the DC operating point of the original full circuit in response to variations

in several of its key design parameters. The new approach leverages, through the

use of circuit moments with respect to the design parameters, the discrete empirical

interpolation approach developed for model reduction in other domains and enables

its efficient application to the problem of DC operating point in nonlinear circuits.

Utilizing the idea of rooted trees, the proposed approach constructs orthogonal bases

that are used in projecting the full equations of the large original nonlinear circuit

onto a reduced system of nonlinear equations in a space with a much smaller dimen-

sion. The variations in the DC operating point of the full circuit are then obtained by

solving the reduced system of equations, yielding significant computational savings.

Numerical examples are presented to demonstrate the efficiency and accuracy of the

reduced model in predicting the change in the DC operating point of the original

circuit.

ii

Acknowledgments

I want to express my deepest gratitude to my supervisor Professor Michel Nakhla.

Throughout my undergraduate studies, he provided me internship opportunities to

introduce me to the world of CAD. He encouraged me to pursue my Master’s degree,

which provided me with the most fulfilling years of engineering studies. His passion

for CAD and learning was an inspiration for me.

I would also like to express my gratitude to my co-supervisor, Professor Emad

Gad. His attention to detail, passion for CAD, and drive were sources of inspiration

throughout my masters. From our routine Skype calls to endless email chains, it was

a once-in-a-lifetime adventure to adjust to the challenges of remote working during

COVID-19.

I want to thank Professor Ram Achar for guiding my fourth-year undergraduate

project. The implementation of Data-Driven Macromodeling (Vector Fitting) on

GPUs was a source of inspiration to pursue CAD and software development further.

Additionally, I would like to thank Ye Tao. It was a pleasure being a Teaching

Assistant with him for the CAD course. As a senior year Ph.D. student in CAD,

he has always been readily available for some friendly deliberations that made my

graduate life more enjoyable. I treasure our friendship.

I am thankful to the staff at the Department of Electronics at Carleton University

for having been so helpful, supportive, and resourceful.

I want to thank my father, Dr. Behzad Nouri, for being a role model throughout

iii

my life. His love for Electrical Engineering, Mathematics, and CAD has always

been an inspiration for me. He has stood by me as an endless source of knowledge

throughout my high school, undergraduate, and now post-graduate journey. I aspire

to be as knowledgeable, passionate, and patient as him as I continue my engineering

career. I would also like to thank my mother for her endless support. She has always

been an invaluable source of encouragement and positivity. I am grateful for my fun

and happy brother Ryan, who is always a steady source of happiness and smiles.

I want to thank Laura Machado for her love and support. She has always en-

couraged me to follow my passion and has shown incredible patience throughout my

masters.

iv

Table of Contents

Abstract ii

Acknowledgments iii

Table of Contents v

List of Tables viii

List of Figures ix

List of Acronyms xi

List of Symbols xii

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objective and Contribution . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Formulation of Nonlinear Systems 5

2.1 Nonlinear Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Nonlinear Circuits as Nonlinear Dynamical Systems . . . . . . 6

2.2 Formulation of Nonlinear Circuits . . . . . . . . . . . . . . . . . . . . 8

2.2.1 MNA Matrix Equations of Nonlinear Circuits . . . . . . . . . 9

v

3 DC Analysis of Nonlinear Circuits 12

3.1 DC Circuit Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Newton-Raphson Iteration to Solve Nonlinear Equations . . . . . . . 13

3.3 Continuation Method for DC Analysis . . . . . . . . . . . . . . . . . 15

3.3.1 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.2 Source Ramping . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Advanced Simulations Using MOR Techniques 19

4.1 MOR in the Context of Circuit Simulation . . . . . . . . . . . . . . . 20

4.2 MOR for Linear Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Construction of Projection Operator . . . . . . . . . . . . . . . . . . 22

4.3.1 Construction of Q Using Krylov-Subspace Methods . . . . . . 22

4.4 MOR for Parametric Circuits . . . . . . . . . . . . . . . . . . . . . . 24

4.4.1 General Formulation of Parametric Circuits . . . . . . . . . . 26

4.4.2 PMOR For DC Solution of Linear Circuits . . . . . . . . . . . 27

4.4.3 PMOR for DC Solution of Nonlinear Circuits . . . . . . . . . 30

4.4.4 The Computational Challenge . . . . . . . . . . . . . . . . . . 32

4.5 Discrete Empirical Interpolation Method (DEIM) . . . . . . . . . . . 33

4.5.1 Reduction of the computational complexity . . . . . . . . . . . 35

4.6 Summary and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 40

5 DC-Centric Parameterized Reduced-Order Model via Moment-

based Interpolation Projection (MIP) Algorithm 42

5.1 Proposed PMOR Approach Using Moment Matching . . . . . . . . . 43

5.1.1 Moment Matching PMOR for DC Operating Point . . . . . . 43

5.1.2 The Main Computations . . . . . . . . . . . . . . . . . . . . . 45

5.2 Proposed MIP-Based Reduction . . . . . . . . . . . . . . . . . . . . . 47

5.2.1 Efficient Projection using MIP . . . . . . . . . . . . . . . . . . 49

vi

5.2.2 Orthonormal Basis U Based on Moment Matching . . . . . . . 51

5.3 Computing the Moments Using Rooted Trees . . . . . . . . . . . . . 52

5.3.1 Computing the Moments Mi(ξ0) and Fi(ξ0) . . . . . . . . . . 53

5.3.2 Construction of the Matrix P . . . . . . . . . . . . . . . . . . 60

5.4 Extension to General Nonlinearity . . . . . . . . . . . . . . . . . . . . 60

6 Numerical Examples 63

6.1 Example 1: CMOS Operational Amplifier . . . . . . . . . . . . . . . 65

6.2 Example 2: 741 Op-Amp . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.3 Example 3: Power Distribution Network (PDN) . . . . . . . . . . . . 71

6.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7 Conclusion 76

7.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.2.1 Example: Low Noise Amplifier . . . . . . . . . . . . . . . . . . 77

List of References 82

Appendix A Fundamental Notions 94

vii

List of Tables

3.1 Scaling the source to solve for the DC solution . . . . . . . . . . . . . 17

6.1 Error at each corner case for the DC simulation of the inverting amplifier. 69

6.2 A comparison between the size of the original system and the reduced

model using PMOR. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.3 A comparison of number nonlinear function evaluation between the

original system and DEIM. . . . . . . . . . . . . . . . . . . . . . . . . 70

6.4 A comparison between the savings achieved during nonlinear function

evaluation using traditional method vs. the proposed method. . . . . 70

6.5 Error at each corner case for the DC simulation of the PDN. . . . . . 73

6.6 A comparison between the size of the original system and the reduced

model using PMOR. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.7 A comparison of number NL function evaluation between the original

system and DEIM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.8 A comparison between the savings achieved during NL function eval-

uation using traditional method vs. the proposed method. . . . . . . 74

viii

List of Figures

2.1 Finite-dimensional dynamical system . . . . . . . . . . . . . . . . . . 5

2.2 A network that accepts inputs and interacts with Peripherals. . . . . 7

2.3 Component Stamps in MNA formulation. . . . . . . . . . . . . . . . . 11

4.1 The subspace of nonlinear function is projecting onto the subspace

spanned by the columns of U. . . . . . . . . . . . . . . . . . . . . . . 36

5.1 An example circuit and its rooted tree representation for JI(x(ξ), ξ). . 58

6.1 Schematic of operational amplifier. . . . . . . . . . . . . . . . . . . . 65

6.2 Variations of the DC voltage at the output node of OpAmp vs. varia-

tions in the length of the MOSFETs channels. . . . . . . . . . . . . . 66

6.3 Variations of the DC voltage at the output node of OpAmp vs. varia-

tions in the width of the MOSFETs channels. . . . . . . . . . . . . . 67

6.4 Schematic of inverting amplifier . . . . . . . . . . . . . . . . . . . . . 68

6.5 Internal schematic of µA741 OpAmp [1]. . . . . . . . . . . . . . . . . 68

6.6 Power distribution network [2]. . . . . . . . . . . . . . . . . . . . . . 71

6.7 Circuit schematic of the SN7404 inverter [3]. . . . . . . . . . . . . . . 72

7.1 LNA schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.2 Steady-state of the output response. . . . . . . . . . . . . . . . . . . . 78

7.3 Comparison between the proposed method and traditional HB for the

variation in the first harmonic of the output node vs. the variation in

the design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 80

ix

7.4 Comparison between the proposed method and traditional HB for the

variation in the second harmonic of the output node vs. the variation

in the design parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 80

7.5 Comparison between the proposed method and traditional HB for the

variation in the third harmonic of the output node vs. the variation in

the design parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 81

x

List of Acronyms

Acronyms Definition

CAD Computer Aided Design

CPU Central Processing Unit

DAE Differential-Algebraic Equation

EIG Eigenvalue (diagonal) Decomposition

EM Electro-Magnetic

FD Frequency Domain

IC Integrated Circuit

I/O Input-Output

KCL Kirchoff’s Current Law

KVL Kirchoff’s Voltage Law

DC-OP DC Operating Point

NL Nonlinear

N-R Newton-Raphson

MOR Model-Order Reduction

ROM Reduced-Order Model

PMOR Parameterized Model-Order Reduction

PDN Power Distribution Network

LNA Low Noise Amplifier

DEIM Discrete Empirical Interpolation Method

SVD Singular Value Decomposition

MPE Missing Point Estimation

EIM Empirical Interpolation Method

TPWL Trajectory Piecewise Linear

RFIC Radio Frequency Integrated Circuits

MIP Moment-based Interpolation Projection

xi

List of Symbols

Symbols Definition

N The field of natural numbers

R The field of real numbers

C The field of complex numbers, e.g.: s-plane

Rn×m The set of real matrices of size n×m

Cn×m The set of complex matrices of size n×m

Cn n differentiable (n-smooth)

a or a∗ The complex conjugate of a complex number a ∈ CAH The complex conjugate of complex matrix A = [aij]

defined as: Aᵀ

= [aji]

Aᵀ

The transpose of matrix A

span The subspace spanned (or generated) by a set of vector

colsp Column space of a matrix

xii

Chapter 1

Introduction

1.1 Background and Motivation

The problem of determining the quiescent (DC) operating point constitutes an in-

dispensable component in the Computer-Aided Design (CAD) tools of integrated

circuits. Typically, this problem requires solving a system of nonlinear equations rep-

resenting the circuit where the difficulty has been traditionally in guaranteeing the

convergence to the DC operating point [4].

Another frequently demanded task in circuit design is to perform the so-called

“parameter sweep” where the DC operating point is reevaluated at numerous values

for several design parameters. Often, the designer considers these parameters as key

to the circuit performance. This demand becomes challenging when the circuit is

large as it requires repeatedly solving a large system of nonlinear equations at the

various values of the design parameters. The cumulative computational cost becomes

prohibitively large as the number of parameters increases.

The general concept of Model-Order Reduction (MOR) emerged as a response to

the need to handle the increasing complexity of the circuits and the mathematical

problems that they spawn in the course of their simulations [5–7]. MOR addresses

the increasing complexity by projecting the full circuit model onto a reduced space.

1

2

The reduced system shares the same essential features as the original (large) system,

allowing it to be simulated in its stead, thereby alleviating its high computational

cost. The idea of Parameterized MOR (PMOR) was born out of MOR to mimic the

original system’s behaviour with respect to the key design parameters identified by

the designer.

Projection-based PMOR methodologies may be grouped by their approach to

constructing the orthogonal basis used in the projection. Moment matching MOR

is one class of projection methods that has been widely adopted in linear circuits

mainly due to its efficiency [8]. Another class of projection-based methods that is

frequently used when the computation of the moments basis is not feasible, such as in

the case of nonlinear systems, is the proper orthogonal decomposition (POD). POD-

based projection is carried out by simulating the full system model to generate the

so-called snapshots used to construct the projection basis.

Moment matching PMOR has been utilized successfully in linear circuits for pur-

poses analogous to the DC operating point (e.g., computing frequency response)

[8–12]. Part of the popularity of moment matching in linear circuits is the low com-

putational cost of constructing the moments projection basis [13,14]. For nonlinear

systems, attempts to apply moment matching PMOR have been made in [15,16]

through approximation of the nonlinear system by a collection of parametric linear

models obtained via linearizing the nonlinearity along training trajectories created

from multiple simulations of the unreduced nonlinear system.

On the other hand, POD-based PMOR can be viewed as a continuation of this

idea but without engaging moment matching [17,18]; it uses the multiple simula-

tions (snapshots) of the full nonlinear model to create the projections basis, thereby

avoiding the need to collect a possibly large ensemble of linear systems.

Obviously, using moment matching PMOR directly on the nonlinear systems still

represents a gap in the literature that, if properly filled, will help in circumventing

3

the need for those snapshots. The goal of this thesis is to fill this gap and use the

proposed approach to address the problem of DC operating point.

1.2 Objective and Contribution

The proposed approach enables using moment matching directly on the nonlinear

system that models the full circuit to construct a reduced-order model that matches

the original circuit’s behaviour with regards to a selected set of design parameters.

The key idea in the proposed approach utilizes the notion of rooted trees to cre-

ate the projection basis. Using the constructed projection basis, and to perform

the projection efficiently, the proposed approach adopts the interpolatory projection

methodology proposed in [19,20] which is known as discrete empirical interpolation

method or (DEIM). The new approach is dubbed as moment-based interpolation pro-

jection (MIP) for its reliance on the moments of the system, as opposed to empirical

snapshots created through simulation, to construct the interpolator operator.

1.3 Organization of Thesis

This thesis is organized as follows:

• Chapter 2 introduces the concept of Modified Nodal Analysis (MNA) to math-

ematically model nonlinear electrical circuits and reviews the strategies to

“stamp” common circuit components in the MNA matrices.

• Chapter 3 presents a common strategy to solve for the DC operating point of

nonlinear circuits. It introduces the application of Newton-Raphson (N-R) as a

method to iteratively find the DC operating point of a nonlinear circuit based

on an initial estimation of the solution. It also discusses common strategies to

help guarantee the solution of N-R, such as “Source Ramping”.

4

• Chapter 4 reviews the application of MOR techniques in the context of cir-

cuit simulation. It highlights the common difficulties and inefficiencies experi-

enced when apply existing MOR techniques to solve nonlinear circuits. It then

presents the Discrete Empirical Interpolation Method (DEIM) to address the

shortcomings of existing MOR techniques.

• Chapter 5 discusses the application of the theory introduced in Chapters 3 and

4 to evaluate the DC solution of nonlinear circuits.

• Chapter 6 presents examples to demonstrate the accuracy and efficiency of the

proposed method.

• Chapter 7 provides closing remarks and discusses possible future work based on

the findings of this thesis.

Chapter2

FormulationofNonlinearSystems

2.1 NonlinearDynamicalSystems

1u

2u

innu

1y

outny

Themainparadigmforlinearsystemsissuperposition,whichisdefinedintermsof

“additivity”and“homogeneity”withrespecttoinputsandinitialconditions. Any

systemthatdoesnotsatisfythesuperpositionpropertyisconsiderednonlinear(NL).

Generalnonlineardynamicalsystems,illustratedinFigure2.1,canbecharacterized

byafinitenumberofnonlinearfirst-orderdifferentialequations,oftenalongwitha

setofalgebraicequations,asgivenin(2.1)[21–23],

Figure2.1:Finite-dimensionaldynamicalsystem

5

6

Σ :

F( t,d

dtx(t), x(t), u(t) ) = 0 (state equations) (2.1a)

y(t) = h( t, x(t), u(t) ) (output equations) (2.1b)

where x(t) ∈ Rn is a vector of system variables, u(t) ∈ Rnin is a vector of input

sources, F ∈ Rn is a vector-valued nonlinear function, y(t) ∈ Rnout is the vector of

the responses at the outputs, and n is the order of system. The mathematical model

in (2.2a) falls in the category of nonlinear Differential-Algebraic Equation (DAE)

systems.

A system is defined as time-invariant if it only consists of time-invariant compo-

nents for which the direct and explicit dependency of equations on the time variable

can be dismissed as

Σ :

F(d

dtx(t), x(t), u(t) ) = 0 (2.2a)

y(t) = h( x(t), u(t) ) (2.2b)

2.1.1 Nonlinear Circuits as Nonlinear Dynamical Systems

A circuit is defined as nonlinear if it consists of at least one nonlinear component (not

counting the independent- voltage and current sources). A circuit element is defined

as nonlinear when the “constitutive relationship” between its voltage (established

across) and its current (flowing through) is a nonlinear function ie = f(ve). A diode

(Id = Is(eVdηVT − 1)) is common example of a nonlinear component.

Electrical circuits are examples of dynamical systems. It was emphasized by Gear

(1968) in [24] that circuits can be well characterized using a system of first-order

differential-algebraic equations (2.2). A complex design is generally comprised of

several circuit elements that are connected together at the nodes. A circuit interacts

with the rest of the design (peripherals) through input/output terminals where the

7

currents enter or exit from the interfacing nodes as shown in Figure 2.2.

node i

node j

node k Sub

-ne

twork

vq-1(t)

v1(t)

u1(t)

u2(t)

up-1(t)

up(t) vq(t)

+

+

+

+

node i

node j

node k Pe

rip

he

ral-

ne

two

rk

v1(t)u1(t)

u2(t)+

+

-++-+

innu (t)

outnv (t)

node i

node j

node k

v1(t)

u1(t)

u2(t)+

+

-++-+

outnv (t)

Pe

rip

he

ral-

ne

two

rk

unin(t)

Figure 2.2: A network that accepts inputs and interacts with Peripherals.

The currents from the sources are added to (or subtracted from) the Kirchhoff’s

Current Law (KCL) equation for the corresponding nodes. The voltage at the ter-

minal nodes (inputs) is directly decided (equated) by the voltage sources at corre-

sponding nodes. The general idea is that the effect of the sources is “linearly in-

jected” [25, 26] into the system at the associated nodes. Being linearly incorporated

in the nodal formulation, the effect of input sources u(t) admits an affine realization

for inputs (c.f. (2.3)).

Following the steps of the nodal analysis [27], the nonlinear state-models in (2.2a)

(for a general class of nonlinear time-invariant circuits) can be recast as [28–30]

d

dtg (x(t)) = F (x(t)) + Bu(t). (2.3)

The following remarks are noteworthy about (2.3)

• The input matrix B is directly applied to the source vector u(t) to map each

source to the connected nodes;

8

• It will be shown in Section 2.2 that under a mild practical assumption, both the

nonlinear differential term at the left-hand side ( ddt

g (x(t))) and the nonlinear

function F (x(t)) can be recast in a simpler form, easing the application of

numerical solvers;

• In the output equation (2.2b), y(t) can be a selection of the voltages and currents

in x(t). The selection can be performed simply by multiplying a selection matrix

L by x(t).

2.2 Formulation of Nonlinear Circuits

There are several established methods to obtain a representation for electrical net-

works, namely,

• Graph-based formulation [31–34],

• Sparse-Tableau Analysis (STA) formulation [33,35],

• Modified Nodal Analysis (MNA) formulations [33,36–38].

One may find a certain method more efficient for specific circuit typologies. The

formulation of the MNA approach is straightforward to implement and is equally

suitable for both frequency and time-domain analyses. Therefore, MNA can be re-

liably used to describe both linear and nonlinear circuits mathematically. Due to

these properties, MNA is the standard formulation method for computer-based cir-

cuit analysis and is commonly used in general-purpose commercial circuit simulators

such as SPICE [39].

MNA is an extension of the nodal analysis method in standard circuit theory [27].

Its strength lies in its ability to handle all types of circuit elements. Constructing

the MNA formulation is usually done on an element-by-element basis and generates

9

a large system containing a mixed set of differential and algebraic equations through

[33,36–38],

• Writing the Kirchoff currents law (KCL) at each node;

• Considering node voltages as the unknowns in the formulation, The circuit

equations will be solved to obtain these unknowns;

• Expressing the currents in the circuit elements in terms of the node voltages

using the admittance form of constitutive relation of the circuit elements;

• Casting a special representation for the elements which either do not have an

explicit representation in an admittance form, e.g., voltage sources or elements

whose constitutive relation requires integration in the time-domain, e.g., induc-

tors;

• Casting the charge or flux based formulation for nonlinear capacitors or nonlin-

ear inductors, respectively, where the pertained charges and fluxs are included

in the unknowns.

2.2.1 MNA Matrix Equations of Nonlinear Circuits

In general, a nonlinear circuit is described using the Modified Nodal Analysis (MNA)

formulation as

Gx(t) + Cd

dtx(t) + f (x(t)) = b(t)

y(t) = Lx(t)

(2.4)

where

• x(t) ∈ Rn×1 is a vector of node voltages appended by currents in inductors,

independent and dependent voltage sources, charges in nonlinear capacitors,

and magnetic flux in nonlinear inductors.

10

• G ∈ Rn×n is a matrix representing the memory-less components of the circuit

such as resistors

• C ∈ Rn×n is a matrix representing the components dependent on the changes

in node voltages (components with memory) such as capacitors.

• f(x(t)) ∈ Rn×1 is a vector of functions describing the nonlinear components. It

contains the currents of nonlinear components such as diodes and the nonlinear

entries corresponding to nonlinear capacitors and inductors.

• b(t) := Bu(t) ∈ Rn×1 is a vector representing the independent voltage and

current sources.

• y(t) ∈ Rnout contains the outputs that are the selections of the voltages and

currents in x(t). The selection can be performed by multiplying x(t) by a

selection matrix L.

• n is the number of variables in the MNA formulation as seen in x(t).

The MNA formulation is constructed on an element-by-element basis. The process

of placing the circuit components in the corresponding MNA matrices (C, G, f(x(t)),

or b(t)) is called “component stamping”. Figure 2.3 shows samples of stamps for

Resistors (R), Capacitors (C), Inductors (L), and Independent Sources.

11

VoltageSource

+ -

SourceCurrent

i

i

i

i

i j

j

j

j

j

J

Element Circuit Schematic

Resistor

Capacitor

Inductor

IE

IL

Stamp

⎡⎢⎢⎣

g · · · −g... · · · ...

−g · · · g

⎤⎥⎥⎦

i ji

j

g

⎡⎢⎢⎣

C · · · −C... · · · ...

−C · · · C

⎤⎥⎥⎦

i ji

j

C

⎡⎢⎢⎢⎢⎣

0 · · · 0 1... · · · ... 00 · · · 0 −11 0 −1 −sL

⎤⎥⎥⎥⎥⎦

i j

i

j

L

⎡⎢⎢⎢⎢⎣

0 · · · 0 1... · · · ... 00 · · · 0 −11 0 −1 0

⎤⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎣

E

⎤⎥⎥⎥⎦

i j

i

j

⎡⎢⎢⎣

J...

−J

⎤⎥⎥⎦

i

j

Figure 2.3: Component Stamps in MNA formulation.(Courtesy of [40])

Chapter 3

DC Analysis of Nonlinear Circuits

A common challenge in simulating an electrical circuit’s behavior using Computer-

Aided Design (CAD) tools is to calculate the DC Operating Point (DC-OP) reliably

and efficiently. As part of this document, I outline the challenges of constructing

a model that tracks the DC-OP variation versus the variations in a selected set of

parameters considered to be key to circuit performance. Before I dive into these

challenges, I must briefly describe the process behind calculating/simulating the DC

operating point.

3.1 DC Circuit Equation

In this chapter, I focus on the steps required to find the DC solution of a general

circuit described by Equation (2.4). Unlike this general form, the DC solution is not

time-dependent. Thus, dx(t)dt

= 0 and x(t) can be denoted simply as x. This simplifies

Equation (2.4) into Equation (3.1). Please note the absence of the C matrix as

multiplying it by dx(t)dt

= 0 eliminates its effects.

Gx + f(x) = b (3.1)

12

13

3.2 Newton-Raphson Iteration to Solve Nonlinear

Equations

The DC solution of Equation (3.1) is defined as finding the values of the vector x

that satisfies

Gx + f(x)− b = 0 (3.2)

In the following, we will use Φ(x) to stand for the residual error that arises from

not satisfying the DC equation for an arbitrary x. In other words Φ(x) is defined by

Φ(x) := Gx + f(x)− b (3.3)

As explained previously, x is the node voltages and the currents of the voltage

sources. Finding the DC solution of Equation (3.1) can be challenging as the equa-

tion’s nonlinear elements (f(x)) depend on the value of x. To circumvent this obstacle,

we can use a numerical method such as Newton-Raphson Iteration.

Newton-Raphson (N-R) is an iterative method that requires the user to make

an initial guess for the value of x. Using this guess, we can compute f(x). At this

point, x and consequently f(x), are an estimation of their true value (“the solution”).

Using this guess as a starting point, the algorithm can iteratively improve the guess

and automatically correct the value of x. Being a numerical method, the number of

iterations that it takes to achieve convergence is dependent on the proximity of the

initial guess to the true solution and the expected accuracy of the solution.

As explained above, for N-R method to satisfy Equation (3.2), it is required to

make an initial guess denoted as x(0). Naturally, the initial guess will likely not be

the correct solution to Equation (3.2). This leaves an error represented by Φ(x(0)),

as seen in Equation (3.4). The next step is to correct x(0) so that the new estimation

14

(x(1)) reduces the error such that Φ(x(1)) < Φ(x(0)).

Φ(x(0)) = Gx(0) + f(x(0))− b (3.4)

To calculate the correction required (∆x(1)) to obtain the new estimation (x(1)), we

use Equation 3.5, where Ψ(x) is the matrix of partial derivatives defined by Equation

(3.6).

∆x(1) = Ψ(x(0))−1Φ(x(0)) (3.5)

Ψ(x) =∂Φ(x)

∂x= G + J(x) (3.6)

J =∂f(x)

∂x(3.7)

Having obtained ∆x(1) from Equation (3.5), we can now modify the initial trial

vector (x(0)) to obtain the next trial vector (x(1)) using Equation (3.8)

x(1) = x(0) −∆x(0) (3.8)

The following process is repeated until convergence is achieved under the following

two conditions, where i represents the ith iteration (assuming i=0 is the initial guess).

i) ‖x(i+1) − x(i)‖ < ε1

ii) ‖Φ(x(i+1))‖ < ε2

In the above two conditions, ε1 and ε2 are predefined error tolerances. Due to

the iterative nature of the N-R method, it is important to choose the initial guess

(x(0)) and error tolerances carefully. Each iteration requires the computationally

involved process of taking the LU decomposition of the Jacobian matrix and For-

ward/Backward substitution. For this reason, the objective should be to minimize

the number of iterations required to achieve an acceptably accurate solution.

15

Algorithm 1 presents a pseudo-code implementation of the N-R method to find

the circuits’ DC solution following the steps described above.

Algorithm 1: Finding DC solution of nonlinear circuits using N-R.

Input: G,b,f(x), x(0) // initial guess

Output: xsol // solution

1

2 Φ(x(0)) = Gx(0) + f(x(0))− b

3 Ψ(x(0)) = G + J(x(0))

4

5 while (‖x(i+1) − x(i)‖ > ε1) or (Φ(x(i)) > ε2) do

// calculate delta x using LU decomposition

6 [L,U,P,Q] = lu(Ψ(x(i−1)))

7 Y = L(P×Φ(x(i−1)))

8 Z = U×Y

9 ∆x(i) = U×Y

10

// calculate new estimation of x

11 x(i) = x(i−1) −∆x(i) // calculate new estimation of X

12

// update using new x estimation

13 Φ(x(i)) = Gx(i) + f(x(i))− b

14 Ψ(x(i)) = G + J(x(i))

15 end

16 return Xsol = X(i)

3.3 Continuation Method for DC Analysis

As previously outlined, a common challenge when using numerical methods such as

N-R is guaranteeing convergence to find the solution. Parameter embedding methods,

also known as Continuation Methods and Homotopy, efficiently address the conver-

gence problems.

This section will go over Homotopy methods and their application in source ramp-

ing to efficiently find the DC solution of nonlinear circuits.

16

3.3.1 Homotopy

Homotopy methods are effective at solving systems of nonlinear algebraic equations.

In the context of circuit simulation, we attempt to tackle a zero finding problem such

as Equation (3.9), where x is Rn and F : Rn → Rn.

F(x) = 0 (3.9)

To find the solution of Equation (3.9), we embed a continuation parameter (λ)

into F(x) to create the Homotopy function H(x, λ) as seen in Equation (3.10).

H(x, λ) = 0 (3.10)

The Homotopy parameter λ ∈ R and the mapping is F : Rn+1 → Rn.

To better explain this, consider Equation (3.11) as an example of a simple Homo-

topy problem.

H(x, λ) = (1− λ)G(x) + λF(x) (3.11)

By setting the continuation parameter (λ) to zero, we get H(x, 0) = G(x) which

has a simple solution. If we set λ = 1, we get H(x, 1) = F(x) = 0 which is our

original problem. Thus, by gradually increasing λ from 0 to 1 and finding the solution

of H(x, λ), we can more reliably calculate the solution of F(x) = 0.

3.3.2 Source Ramping

Source ramping is a special use-case of Homotopy to increase the probability of con-

vergence. As the name implies, source ramping requires the gradual increase of the

DC source voltages from zero using a multiplication factor (α). Equation (3.12)

demonstrates the incorporation of the scaling factor in the general nonlinear MNA

17

equation.

Φ(x, α) = Gx + f(x)− αb = 0 (3.12)

As explained in Section 3.2, the first step of the process is to set α to an arbitrarily

small number such 0.2 and set the initial guess as zero (x(0)1 = 0). N-R is then used

to find the solution which takes ν iterations denoted as x(ν) = x(sol). Now that the

solution is found, the DC source voltages are gradually scaled up in stages, and the

solution from the previous stage is used as the initial guess of the next stage. This

process is illustrated in Table 3.1.

Table 3.1: Scaling the source to solve for the DC solution

Source Ramping Index Source Scale (α) Initial Guess Solution

1 0.2 x(0)1 = 0 x

(sol)1

2 0.4 x(0)2 = x

(sol)1 x

(sol)2

3 0.6 x(0)3 = x

(sol)2 x

(sol)3

4 0.8 x(0)4 = x

(sol)3 x

(sol)4

DC Solution 1 x(0)5 = x

(sol)4 x

(sol)5

As explained previously, N-R is an iterative method that can take several iterations

to converge. However, convergence is not always guaranteed. By implementing source

ramping, it increases the probability of convergence. However, with each gradual

increase in α, the entire N-R process has to be rerun, making this method even more

computationally demanding than N-R alone. That is why it is important to carefully

choose the step size at which α increases to guarantee convergence but minimizes the

computational complexity (total number of N-R iterations).

Evidently, optimizing the computational cost and reducing simulation times is

important when tackling circuit simulation problems. With the combination N-R

and source ramping, the total number of N-R iterations rapidly balloons. Chapter 4

18

will introduce methods to reduce the computational cost of each N-R iteration. This

optimization will have a favourable effect on reducing the overall computational cost

of DC circuit simulation.

Chapter 4

Advanced Simulations Using MOR

Techniques

Model-order reduction (MOR) has proven to be an effective tool in reducing the

computational complexity of simulating large systems. MOR has been successfully

used in a broad spectrum of linear and nonlinear applications [41–48] such as micro-

electronics [39,40,49–51], high-speed and RF circuits [52–55], uncertainty quantifica-

tion [9,56–58], electromagnetic [59,60] and thermal analysis [61]. The recent evolution

of MOR techniques have also been fueled by their popularity and success in broader

fields such as mechanical, biomedical, civil, and aerospace engineering [62–64].

This chapter is intended as the primary platform for reviewing the notations

related to Model-Order Reduction (MOR) and its application in circuit simulation.

Naturally, the concepts, techniques, and applications of MOR are too vast to be

covered in one chapter. Rather, this chapter is more focused on the techniques that

form the basis for the contribution of this thesis. The organization of the chapter

proceeds along the following sections.

Section 4.1 introduces the idea of MOR as a general concept, while Section 4.2

shows its application in the domain of linear circuits through the concept of projection

based MOR. Section 4.3 then presents a more detailed look in the computation of

19

20

the projection operator. Section 4.4 then moves to the more specialized idea of

Parameterized MOR (PMOR) which is the principal technique used in this thesis.

The presentation of this section reviews the application of PMOR in linear circuits.

It subsequently explores the application of PMOR in general nonlinear circuits to

highlight the main challenges which are addressed by the contribution in this thesis.

4.1 MOR in the Context of Circuit Simulation

Model-Order Reduction is an effective technique used to reduce the computation time

required to simulate large circuits. The high-level strategy behind MOR techniques

is to speed up the simulation time by reducing the order of the circuit’s mathematical

model. By reducing the order of the model, it is possible to reduce the computation

time but at the risk of reduced accuracy. However, in most applications, the model’s

order can be reduced to a certain extent to minimize the CPU-cost without noticeable

degradation of accuracy.

4.2 MOR for Linear Circuits

Model-Order Reduction (MOR) for linear systems is a well-established area, and a

rich body of literature is available on the subject [43,46,51,65–67]. This section will

discuss a common approach to reducing linear systems based on using orthogonal

projection operators.

As previously explained, the MNA equation for a general linear circuit is obtained

by dismissing the nonlinearity from (2.4) to get (4.1).

Gx(t) + Cdx(t)

dt= Bu(t)

y(t) = Lx(t)

(4.1)

21

where G,C ∈ Rn×n, B ∈ Rn×nin , u(t) is the circuit’s nin stimuli at the inputs,

L ∈ Rnout×n, and n is the size of the original system.

The key idea behind MOR is to reduce the number of state variables by projecting

the vector x(t) on to an m-dimensional subspace spanned by the column vectors of

the orthogonal matrix Q ∈ Rn×m as

x(t) = Qx(t) (4.2)

where x(t) ∈ Rm×1 and m n. Next, using (4.2), the Reduced Order Model (ROM)

is obtained through the Galerkin projection scheme [48] as

Gx(t) + Cdx(t)

dt= Bu(t)

y(t) = Lx(t)

(4.3a)

where

C∆= Q>CQ ∈ Rm×m B

∆= Q>B ∈ Rm×nin

G∆= Q>GQ ∈ Rm×m L

∆= LQ ∈ Rnout×m (4.3b)

By preserving specific properties in the original system, the reduced model in

(4.3a) should ideally provide a sufficiently accurate approximation for the system

response x.

The framework introduced above for MOR leaves out the details of the construc-

tion of the projection matrix Q or the requirements that it needs to satisfy in order

for the reduced system (4.3a) to preserve the essential features of the original system

(4.1) and guarantee its accuracy. The computation of Q will be discussed in more

depth in Section 4.3.

22

4.3 Construction of Projection Operator

Using the projection framework described above as an effective reduction strategy

has been extensively explored in literature for linear [51, 68–74] and nonlinear [44,

50, 75–84] systems. The key steps in the projection operation is the construction of

the projection operator matrix Q. The method employed to obtain Q is the main

differentiating factor between the reduction methods.

Finding suitable projection matrices is the main task in any MOR method. There-

fore, the next section will elaborate on the construction of Q as related to the subject

of this thesis.

4.3.1 Construction of Q Using Krylov-Subspace Methods

The Krylov-Subspace approach to constructing the projection matrix for use in the

model-order reduction of linear circuits necessitates working in the Laplace-domain

instead of time-domain.

By applying Laplace transformation to the MNA equations in (4.1), the corre-

sponding complex-valued matrix transfer function (in complex frequency s-domain) is

obtained as a relationship between the Laplace-domain output Y(s) and the Laplace-

domain input U(s), in the form

Y(s) = H(s)U(s) (4.4)

where

H(s) = L (G + sC)−1 B, (4.5)

It is reasonable to assume that the matrix pencil (G,C) is “regular” which means

that the matrix (G + sC) is non singular except at a finite number of points in the

Laplace-domain. With this assumption, a properly selected value so is chosen in the

23

Laplace-domain such that (G + soC) is non-singular. Using a properly selected so as

an expansion point, the transfer function (4.5) can be expanded in Taylor series in

the vicinity of so as

H(s) = L∑

j=0

Mj(so)(s− so)j (4.6)

where the coefficient of (s− so)j,

Mj(so) =1

j!

∂jH(s)

∂sj

∣∣∣s=so

is called the jth moment of the system’s variables at so and is given by [51]

Mj(so)∆= Aj R (4.7a)

where

R∆= (G + soC)−1 B ∈ Rn×nin (4.7b)

A∆= − (G + soC)−1 C ∈ Rn×n . (4.7c)

The moments for the transfer function can be obtained using the system’s moments

in (4.7) as

Mi(so) = LMj(so) ∈ Rnout×nin (4.8)

The so-called Krylov-Subspace MOR methods constructs the orthogonal projec-

tion matrix by ensuring that

1. The m columns of Q are orthonormal, i.e.,

Q>Q = Im (4.9)

24

2. The m-dimensional space of the columns of Q span the space formed by the

columns of R, AR, . . . , A(m−1). This feature of Q is typically expressed by

colsp Q = Kr(A,R,m) = spanR, AR, . . . , A(m−1)R

(4.10)

where m = j × nin, j is the number of the block-moments, and nin is the number

of the columns of B in the circuit formulation. The Arnoldi algorithm is an example

of an efficient method for computing Q [85].

Once Q is constructed, it is then used to construct the reduced system as shown

in (4.2) and (4.3).

Through Galerkin projection [65, 86] and using the projection matrix Q which

satisfies the above two conditions, the reduced order model (4.3a) and the original

(4.1) will share the first m moments about the expansion frequency point so. This

is key to ensure that the reduced system will approximate the original system with

good accuracy [49,69].

4.4 MOR for Parametric Circuits

The term “parametric circuits” is used in this thesis to identify circuits with several

design parameters that have been designated as key or critical to the performance of

the circuit with respect to certain metrics.

During the circuit design phase, specific design parameters (e.g., layout and ma-

terial features) influence the circuit’s behaviour and must be further analyzed by

designers to satisfy the design specifications. This leads to design tasks such as de-

sign space exploration, optimization, and variability analysis that are, in general,

computationally cumbersome. The high computational complexity arises from hav-

ing to repeatedly run the simulation of the circuit at numerous points in the space of

25

the design parameters.

The objective of this section is to review the role that MOR techniques play in

reducing the computational cost associated with the idea of design space explorations.

In this context, MOR is often referred to as Parameterized MOR (PMOR) to empha-

sise the fact that the obtained reduced system approximates the original system in

regards to its behaviour with respect to a selected set of design parameters. The term

“design space exploration” is also known under different names, such as the “param-

eter sweep” which is used in circuit simulation to refer to the notion of repeating the

simulation under different values for the selected set of design parameters. Several

PMOR methods have been developed for linear [8–10, 12, 59, 87–95] and nonlinear

systems [15, 16, 96, 97] to capitalize on the savings achieved by using reduced order

models.

The plan for this review proceeds along the following lines:

1. First, the mathematical formulation of the general nonlinear circuit (MNA) is

modified to account for the presence of the design parameters whose effect on

the circuit performance needs to be explored. This is done in section 4.4.1.

2. An overview of the PMOR approach that is proposed to handle the task of

design space exploration in linear circuits is presented in Section 4.4.2.

3. Section 4.4.2 will push the idea of PMOR to the domain of nonlinear circuits

using the same scheme employed in linear circuits. It will be shown that this

attempt will engender new challenges not seen in the application of PMOR

in linear circuits. More particularly, using the linear-type of PMOR directly

on the nonlinear circuit will force the computation to alternate between the

reduced-dimension space and the original large dimensional space of the cir-

cuit, creating a computational bottleneck whose resolution will be presented

26

in the following section (section 4.5) through the idea of Discrete Empirical

Interpolation Method (DEIM).

4.4.1 General Formulation of Parametric Circuits

The dependence of a circuit’s response on the design parameters ξ = ξ1, . . . , ξd ∈Ω ⊂ Rd can be represented by adapting the modified nodal analysis (MNA) formu-

lation to the following form

G(ξ)x(t, ξ) + C(ξ)dx(t, ξ)

dt+ f(x(t, ξ), ξ) = Bu(t)

y(t, ξ) = Lx(t, ξ)

(4.11)

where C(ξ),G(ξ) are n× n parameter-dependent conductance and susceptance ma-

trices, respectively. x(t, ξ) ∈ Rn is the vector of parameter-dependent MNA variables,

y(t, ξ) contains the nout parameter-dependent output response, B ∈ −1, 0, 1n×nin ,

and L ∈ −1, 0.1nout×n are input and output selector matrices. The vector-valued

function f(x(t, ξ), ξ) represents the nonlinearity in the circuits with a nonlinear de-

pendence on both the parameters ξ ∈ Ω and parameter-dependent MNA variables

x(t, ξ). The parametric linear circuits can be represented by dismissing the nonlinear

term from (4.11).

Numerous design tasks, such as design optimization and sensitivity and variability

analysis, require multiple simulations of the system for multiple variations in the

design parameters (e.g., layout features of an electronic system). Simulating the

original large-scale models of these systems (4.11) for each parameter variation can

become computationally expensive.

The main ideas of PMOR will be introduced through its applications in computing

the variation of the DC operating point with regards to variations in the design pa-

rameters ξ. PMOR operates in the same framework of the MOR approach: projecting

27

the original system matrices onto a reduced space using a projection operator. How-

ever, the reduced system matrices of PMOR preserve the dependence of the system of

equations on the parameters. As a result, an approximation of such dependencies is

manifested in the reduced model. There are certain strategies in literature to achieve

this goal [12].

4.4.2 PMOR For DC Solution of Linear Circuits

The problem formulation for computing the DC operating point of linear circuits is

obtained by dismissing the nonlinear f(x(t, ξ), ξ) and the derivative terms from 4.11

as

Φdc(xdc(ξ), ξ) := G(ξ)xdc(ξ)− bdc = 0 (4.12)

where xdc(ξ) ∈ RN is the DC solution, ξ ∈ Ω is a vector collecting d circuit parameters

ξ1, . . . , ξd, bdc ∈ RN is a vector containing the DC sources in the circuit, and N is

the number of circuit variables.

The purpose of solving parametric linear systems, as given in (4.12), is to track

the variation of the DC operating point xdc(ξ) of a circuit in response to variation

in parameters ξ. The large systems defining today’s complex designs need to be

repetitively solved for different values of the parameters.

The central idea in using PMOR is to approximate the large system in (4.12) with

a reduced model, while preserving the same dependency on the parameters for the

reduced solution. Tracking the variation of the DC operating point xdc(ξ) can be

performed by solving for the solution in the reduced space. This leads to a noticeable

reduction in CPU-cost.

PMOR based on moment matching has been applied for linear circuits for purposes

analogous to the DC operating point (e.g., computing the variation in the frequency

response) [8–12]. In the context of DC operating points, moment-matching based

28

PMOR involves finding a reduced DC equation whose solution has a few leading

parameter moments that match those of the original DC solution (4.12).

Single Parameter Moment-Matching PMOR for DC Operating Point of

Linear Circuits:

In order to find the parametric reduced order model, the moments of the DC response

of the circuit with respect to the random parameters are obtained.

To facilitate the presentation of the idea and without loss of generality, a single

design parameter of interest is assumed in the following derivation (i.e., d = 1 and

the vector ξ is replaced with the scalar ξ).

This approach proceeds by first expressing the set of variables xdc(ξ) as a Taylor

series around the nominal value of the parameter, denoted ξ0

xdc(ξ) =∞∑

i=0

Mi(ξ0) (ξ − ξ0)i (4.13)

where

Mi(ξ0) :=1

i!

∂ixdc(ξ)

∂ξi

∣∣∣∣ξ=ξ0

∈ Rn (4.14)

are the so-called moments. Forming the reduced system is carried out by first con-

structing an orthogonal basis set Q ∈ Rn×m whose column vectors form a basis for

the subspace spanned by the moments.

colsp Q = span M0(ξ0),M1(ξ0), · · ·Mm−1(ξ0) (4.15)

For the sake of illustration, assume that G(ξ) is dependent on ξ in an affine manner,

that is

G(ξ) = G(ξ0) + ξG(ξ0). (4.16)

29

Next, the reduced system is obtained from the original system 4.12 using variable

change as

xdc(ξ) = Q xdc(ξ) (4.17a)

and through congruent transformation,

Φ(xdc(ξ), ξ) = G(ξ) xdc(ξ)− bdc = 0 (4.17b)

where

G(ξ)∆= Q>G(ξ0)Q + ξQ>G(ξ0)Q, bdc

∆= Q>bdc. (4.17c)

Multi-Parameter Moment-Matching PMOR for DC Operating Point of

Linear Circuits:

Similar to the single parameter case, the development of multi-parameter moment-

matching PMOR to find the DC operating point of nonlinear circuits is established

based on the idea of expanding xdc(ξ) in the Taylor series around an expansion point

ξ0 = ξ1,0, . . . , ξd,0, which is the vector of nominal values for the parameters. A

multidimensional representation of the Taylor expansion of x(ξ) using the notion of

multi-index α ∈ Nd is given as

xdc(ξ) =∑

αMα(ξ0)

d∏

i=1

(ξi − ξi,0)αi (4.18)

where Mα(ξ0) are the coefficients of the DC solution’s Taylor series and α in this case

is referred to as a multi-index (a vector of indices). Let the Taylor series expansion

(4.18) be truncated such that α is confined to a set signified as Λ. More details about

(4.18) will be discussed in the next sections (c.f. Section 4.4.3).

Next, a set of orthogonal basis spanning the space defined by the columns of the

30

moments is computed [10,12,98]

colsp Q = spanMα(ξ0), α ∈ Λ (4.19)

where Q ∈ Rn×m and m is the number of elements in Λ (m n).

Also, by an affine representation for G(ξ), as proposed in [10], we have

G(ξ) =d∑

i=1

Giξi. (4.20)

Using the orthogonal matrix Q from (4.19), the ROM can be computed following the

projection scheme as

xdc(ξ) = Qxdc(ξ). (4.21)

In a similar fashion, as shown in (4.17), the parametrized ROM for the DC solution of

multi-parameter systems can be written through congruent transformation and using

(4.20) as

Φdc(xdc(ξ), ξ) = G(ξ)xdc(ξ)− bdc = 0 (4.22a)

where

G(ξ)∆=

d∑

i=1

Q>GiQ, bdc∆= Q>bdc. (4.22b)

4.4.3 PMOR for DC Solution of Nonlinear Circuits

Similar to the linear circuits, the problem formulation for computing the DC operating

point of nonlinear systems (4.11) is obtained by

Φdc(xdc(ξ), ξ) := G(ξ)xdc(ξ) + f(xdc(ξ), ξ)− bdc = 0 (4.23)

31

Applying PMOR on nonlinear circuits using moment matching has not been ad-

dressed before in the literature. The objective in this section is to attempt expanding

the PMOR technique, as it is used in linear circuits through moment matching, to

nonlinear circuits.

The steps for the reduction of (4.23) based on moment matching and projec-

tion are illustrated below to highlight the associated challenges and computational

bottlenecks. These issues will be discussed in detail in the upcoming sections.

Applying the moment-matching approach used to reduce the above mentioned

system (4.23) requires the following main steps,

1. Expanding x(ξ) in Taylor series around ξ0 with a multidimensional representa-

tion by using the notion of multi-index with α ∈ Nd as

xdc(ξ) =∑

αMα(ξ0)

d∏

i=1

(ξi − ξi,0)αi (4.24)

where Mα(ξ0) are the moments of the truncated Taylor expansion.

2. Forming an orthonormal basis Q ∈ RN×m for the subspace spanned by the set

of moments collected from the truncation to the set Λ,

colsp Q = span Mα(ξ0) , α ∈ Λ (4.25)

3. Finally, using an affine representation for G(ξ), as proposed in [10],

G(ξ) =d∑

i=1

Giξi, (4.26)

premultiplying (5.1) by Q> and through the change of variables,

xdc(ξ) = Qxdc(ξ) (4.27)

32

a reduced system of the form

Φ(xdc(ξ), ξ) := G(ξ) xdc(ξ) + f(xdc(ξ), ξ)− bdc, (4.28)

is obtained, where

G(ξ) =d∑

i=1

Q>GiQξi, (4.29)

f(xdc(ξ), ξ) = Q>f(xdc(ξ), ξ), (4.30)

bdc = Q>bdc (4.31)

The above steps show that in the course of using PMOR for tracking the variations

in of the DC operating point for values of ξ different than the nominal values ξ0, one

would solve the reduced system Φ (xdc(ξ), ξ) = 0 for xdc(ξ) at those values of ξ and

use Q to map xdc back to xdc and obtain the corresponding DC operating point of the

original system, xdc(ξ) = Qxdc(ξ). Nonetheless, this simplistic take on the problem

would incur computational difficulties that are discussed next.

4.4.4 The Computational Challenge

The above section highlighted two main issues that emerged as essential to applying

the moment matching PMOR in the context of DC operating point. The first issue is

the computation of the orthogonal basis Q that will be used in the projection. The

second issue arises in the course projecting the nonlinear term in (4.30) in the reduced

space. The rest of this section takes a closer look into the requirements needed in the

implementation of the second issue, leaving the first issue for Section 5.3.

Solving for the reduced system (4.28) for xdc(ξ) is typically done through ap-

plying the Newton-Raphson iterative scheme. The core of this process requires the

33

computation of the reduced nonlinear function and its Jacobian matrix, which are,

respectively, obtained from,

f(xdc(ξ), ξ) = Q>f(Qxdc(ξ), ξ) (4.32)

and

J(xdc(ξ), ξ) =∂ f(xdc(ξ), ξ)

∂xdc

= Q>J(Qxdc(ξ), ξ)Q (4.33)

Using the values obtained for xdc(ξ), it is possible to recover the values in xdc(ξ)

through the mapping Qxdc(ξ).

It is obvious from the procedure, as outlined above, that the main advantage

of using the proposed approach to track the variation of the DC operating point is

having to factorize the reduced Jacobian matrix with size m, instead of the original

one which of size N and given that typically m << N .

However, the basic computation of the reduced nonlinear f ∈ Rm and J ∈ Rm×m

(as presented in (4.32) and (4.33), respectively) would still require invoking the full

circuit device model evaluation routines to compute its contribution to the N com-

ponents of f(·) as well as the nonlinear devices stamp on the N ×N original Jacobian

matrix J(·). This, of course, comes in addition to the O(mN) computational com-

plexity incurred in the linear algebra operations involving the projection Q. This

fact represents a significant disadvantage that could curb the computational savings

hoped for by entirely moving the computation from RN to the Rm domain.

4.5 Discrete Empirical Interpolation Method

(DEIM)

As explained in Sections 4.1–4.2, model-order reduction (MOR), and by extension

PMOR, have proven to be an effective tool in reducing the computational cost of

34

large linear circuits.

However, the application of MOR for nonlinear circuits remains a challenging task.

As elaborated in Section 4.4.4 for the case of Nonlinear DC problems, the application

of moment matching PMOR suffers two main issues that can easily outweigh any

advantage of the reduction.

These issues are extremely important and are the main barriers in the applica-

tion of projection based reduction even for the general case of nonlinear systems.

Hence, when the MNA system of equations contains nonlinear components (f(x)),

the computational savings achieved by the projection-based MOR is not as evident.

Although the dimension of the reduced nonlinear circuits (4.28) is smaller than the

original (unreduced) system (5.1), the computational complexity for the evaluation

of the reduced nonlinear function f(x) and its Jacobian J(x) does not improve. The

computation of these reduced functions still requires evaluating f(x) and its Jacobian

J(x) in the unreduced domain using the original system. It also requires taking x

back and forth from the reduced space to the original space to evaluate f(x) and

J(x). Once the computational investment has been made to evaluate each individual

element of f(x) and J(x) only then can they be reduced. This makes the simulation

of the reduced-order system inefficient.

There have been several attempts in the literature to circumvent this problem.

Historically, the first was to replace (approximate) the nonlinearity by a weighted

combination of linear systems which could then be efficiently treated by well-known

linear reduction methods. The trajectory piecewise-based techniques in [75,76,99,100]

are developed using this approach. For a more detailed insight one can refer to [101]

and references therein.

Alternatively, there have been several recent papers attempting to address this

issue such as Gappy-POD [102], Missing Point Estimation (MPE) [45,103], Empirical

Interpolation Method (EIM) [56, 104], and its recent discrete variant called Discrete

35

Empirical Interpolation Method (DEIM) [19,52,105,106].

This section will outline the general formulation of DEIM. DEIM aims to achieve

an approximation for f(x) and J(x) based on combining projection and interpolation

approaches. To this end, some selected k components of the nonlinear function,

denoted fk(xk(t)) (k n), are computed and used to approximate the full nonlinear

vector.

4.5.1 Reduction of the computational complexity

An orthogonal basis U ∈ Rn×k that spans the subspace of nonlinear function F is

constructed. In the classical DEIM, also known as POD-DEIM, F is defined by the

snapshots of the nonlinear function as

F = [f(x(t1)), f(x(t2)), . . . , f(x(tN))] ∈ Rn×N . (4.34)

where f(x(ti)) ∈ Rn is the evaluation of the nonlinear vector at time instance ti and

N is the number of time samples. These N samples can be collected, without any

extra computational cost, from the initial simulation of the circuit which is required

by the conventional POD reduction project operator Q ∈ Rn×m (m n).

The orthogonal basis UF ∈ Rn×k is obtained through a Singular Value Decompo-

sition (SVD) [107] of the nonlinear data matrix F as,

F = UFΣFWFᵀ

(4.35)

This results in N left-singular vectors in the orthonormal matrix UF ∈ Rn×N which

are the basis spanning the space of the nonlinear trajectory data, i.e.,

UF = colspan(F) (4.36)

36

and ΣF = diag [σ1, . . . , σN ] RN×N is the singular-value matrix, where σ1 ≥ σ2 ≥. . . ≥ σN ≥ 0, let (N < n).

The projection operator U (4.37) is constructed using the first k left-singular

vectors in UF corresponding to the k largest singular values in ΣF .

U = ui ∈ UF | for i = 1, . . . , k; k n ∈ Rn×k (4.37)

An approximation (c(t)) for the nonlinear function f(x(t)) can be obtained by pro-

jecting it onto the subspace spanned by the columns of U, as illustrated in Figure 4.1

and shown in (4.38).

Projection

f(x(t))=Uc(t)

f c

Figure 4.1: The subspace of nonlinear function is projecting onto the subspacespanned by the columns of U.

f(x(t)) ≈ Uc(t) (4.38)

A selector matrix P ∈ Rn×k is constructed in Algorithm 2. Using this algorithm,

the k rows of the nonlinear function that have the most dominant contribution to the

nonlinear space, spanned by the column of U, are captured. Matrix P is formed as

37

P = [eγ1 , . . . , eγk ] ∈ 0, 1n×k. (4.39)

where eγi is the γi-th column of an n× n identity matrix.

As a result, fk(xk) from (4.40) includes the most significant k rows (components)

in f(x)

fk(xk) := P>f(x) (4.40)

fr(xr(t)) =[fγi(xfγi

(t)) | for γi ∈ γ1, . . . , γk]∈ Rk×1 (4.41)

where xfγi

(t) ⊆ x(t). The vector xfγi

(t) collects the entries in unknown vector x(t)

that are needed for the evaluation of the functions contributing to the γi-th row in

f(t). Therefore, the truncated vector of unknowns xk(t) can be defined as a collection

of those entries in x(t) that are needed for the evaluation of all the rows in fk. This

can be formalized as

xk(t) = xfγ1 (t)⋃

xfγ2 (t)⋃· · ·

⋃xfγk (t) = Υ>x(t) ∈ Rη (4.42)

where η is the number of the MNA variables in xk(t) and Υ ∈ 0, 1n×η is a selector

matrix selecting the entries from x(t) required to compute fk.

Thus, the approximated nonlinear vector can be represented as below,

fk(Υ>x(t)) = P>Uc(t) (4.43)

Using (4.38), (4.43), and the projection Q for reduction (e.g., form POD method),

an approximation for the reduced nonlinear function can be obtained as

f(x(t)) ≈ Ψfk(Υ>Qx) (4.44)

38

where Ψ is defined as,

Ψ = Q>U(P>U)−1 ∈ Rm×k (4.45)

A similar technique can also be used to approximate the Jacobian as seen in Equation

(4.46).

J(x) ≈ ΨJk(xk)Υ>Q ∈ Rm×m (4.46)

where

Jk(xk)∆=∂fk(xk)

∂xk= P>J(x)Υ ∈ Rk×γ. (4.47)

Evidently, the evaluation of the reduced nonlinear function and its Jacobian are per-

formed in the reduced domain. This reduces the computational complexity for the

evaluation of f(x) from O(n) to O(k) where (k n). This translates to a significant

reduction of CPU-cost.

39

Algorithm 2: Computation of the selector matrix P.

Input: F ∈ Rn×N , (Nonlinear data matrix)

m, (The prescribed reduction order)

K, (The order for interpolation space)

Output: U ∈ Rn×k, (Interpolation projector)

P ∈ 0, 1n×k, (Row selector matrix)

Γ ∈ Nk (Matrix of interpolation indices)

1: (ui, σi)← svd(F); for i = 1, . . . ,m2: [ρ, γ1] = max(|u1|); where ρ ← |u1|∞ and γ1 ← Row# of ρ3: In ← eye(n, n); (Identity matrix of order n)

4: U← u1;5: eγ1 ← In(:, γ1); (γ1-th column in identity matrix)

6: P← eγ1 ;7: Γ← γ1;8: k ← 1;9: while k < K do10: k ← k + 1;11: Solve (PtU)c = Ptuk for c;12: r = uk −Uc;13: [ρ, γk] = max|r|;14: U← [U,uk];15: P← [P, eγk ];16: Γ← [Γ, γk];17: end while18: return U, P, Γ;

40

4.6 Summary and Discussions

The main objective in this chapter has been to outline the concepts of MOR and

its application in the context of parameterized MOR (PMOR). The underlying goal

of this presentation has been the creation of reduced system of nonlinear equations

whose solution tracks the variations of the DC solution of the circuit with respect to

variation in some design parameters. The approach used to achieve this goal relied

on extending the concepts that are used in linear PMOR to the nonlinear domain.

Although the procedural steps are well-defined and succeed in creating a reduced

system with the desired characteristics, the development of those steps in the chapter

exposed several issues and drawbacks that need to be addressed. More specifically,

the contribution of this thesis is with regards to the following issues:

1. Computation of the orthogonal basis Q that is used in the projection of the

original system onto the reduced space. For the case of linear circuits, this issue

is handled through a well established approach that was described in Section 4.3.

This approach, however, cannot be used for nonlinear circuits. The resolution

of this issue is part of the contribution in this thesis and will be presented in

the next chapter.

2. Projecting the nonlinear part of the DC nonlinear equations, f(x) on the re-

duced space. The development in this chapter demonstrated that the projec-

tion is a bottleneck in itself, since unlike the linear case, the projection needs

to be repeated each time the reduced system needs to be solved. The DEIM

framework which was presented in Section 4.5 was proposed to solve that pro-

jection problem but in different domains, e.g. aerodynamics and fluid mechan-

ics [20,108,109]. DEIM, however, as it is implemented in those domains, requires

using the POD via snapshots of f(x). This is not suitable in the context of DC

problems since it forces solving the original DC problem in order to generate the

41

snapshots. The contributions of this thesis addresses this issue in using DEIM

in the projection operation.

The following chapter presents the contributions of the thesis which handle the

above-mentioned issues.

Chapter 5

DC-Centric Parameterized

Reduced-Order Model via Moment-based

Interpolation Projection (MIP) Algorithm

This chapter presents the main contributions of the thesis. The contribution in-

troduced by this thesis targets the application of PMOR in nonlinear circuits as a

computationally efficient tool to track the variations in the DC operating point of the

circuit in response to variations in a selected set of design parameters.

The proposed approach uses the idea of direct moment matching to construct a

reduced system that is used in tracking the DC solution of the original system.

The presentation style used in this chapter is top-to-bottom in the sense that the

main ideas are first presented. Through this presentation, key building blocks are

identified and detailed computational steps are presented next.

42

43

5.1 Proposed PMOR Approach Using Moment

Matching

This section describes the main outline of the proposed approach. Through the

developments presented in this section, the essential procedural steps involved will be

highlighted. Further details on the implementation of those steps will be handled in

the following sections.

The mathematical formulation for a general nonlinear circuit with d design pa-

rameters ξi, i = 1, · · · , d is typically obtained through the modified nodal (MNA)

analysis approach [36]. Using MNA yields a system of nonlinear algebraic equations

of the form

Φ(x(ξ), ξ) := G(ξ)x(ξ) + f(x(ξ), ξ)− b (5.1)

where G(ξ) ∈ RN×N is a matrix that carries the stamps of memoryless linear

elements, f(x(ξ), ξ) ∈ RN is a vector of N nonlinear functions that describe the

nonlinear elements. b ∈ RN is a vector containing the quiescent (DC) sources in the

circuit, N is the number of circuit variables, and ξ = [ξ1, · · · , ξd]>.

The task of finding the DC operating point for a specific (nominal) set of parame-

ters, say ξ = ξ0, is done by using an iterative technique such as the Newton-Raphson

to solve Φ(x(ξ0), ξ0) = 0.

5.1.1 Moment Matching PMOR for DC Operating Point

PMOR based on the concept of moment matching has been applied for linear circuits

for purposes analogous to the DC operating point (e.g., frequency response) [8–12].

For nonlinear circuits, however, using PMOR through moment matching on the orig-

inal nonlinear system of equations that models the circuit has not been addressed

44

before. The proposed approach handles the issues involved in this objective and uses

the resulting algorithm to efficiently track the variations in the DC operating point

in response to variations in key design parameters.

The proposed approach proceeds conceptually in the following four main steps,

1. Expand x(ξ) in Taylor series around ξ0. Typically, representing a multidimen-

sional Taylor expansion of x(ξ) is better served by using the notion of multi-

index, which defines a multi-index, say α, of size d as a vector with non-negative

elements, i.e., α ∈ Nd. Thus, a Taylor expansion for x(ξ), is represented by

x(ξ) =∑

αMα(ξ0)

d∏

i=1

(ξi − ξi,0)αi (5.2)

where Mα(ξ0) are the Taylor series coefficients, which are also known as the

moments of the expansion,

2. Truncate the Taylor expansion to include only α in a set, denoted Λ, defined

according a certain criterion. For example, using the notion of total order, Λ

is given by Λ := α : Σiαi < p for some integer p. In the following we shall

assume that Λ includes m elements (|Λ| = m) with m << N .

3. Forming an orthonormal basis Q ∈ RN×m for the subspace spanned by the set

of moments collected from the truncation,

Q = colspan [Mα(ξ0)] , α ∈ Λ (5.3)

4. Finally, using an affine representation for G(ξ), as proposed in [10],

G(ξ) =d∑

i=1

Giξi, (5.4)

45

premultiplying (5.1) by Q> and through the change of variables,

x(ξ) = Qx(ξ) (5.5)

a reduced system of the form

Φ(x(ξ), ξ) := G(ξ) x(ξ) + f(x(ξ), ξ)− b, (5.6)

is obtained, where

G(ξ) =d∑

i=1

Q>GiQξi, (5.7)

f(x(ξ), ξ) = Q>f(x(ξ), ξ), (5.8)

b = Q>b (5.9)

The above steps show that in the course of using PMOR for tracking the variations

in of the DC operating point for values of ξ different than the nominal values ξ0, one

would solve the reduced system Φ (x(ξ), ξ) = 0 for x(ξ) at those values of ξ and use

Q to map x back to x and obtain the corresponding DC operating point of the original

system, x(ξ) = Qx(ξ). Nonetheless, this simplistic take on the problem would incur

computational difficulties that are discussed next.

5.1.2 The Main Computations

The presentation of the previous subsection highlighted two main issues that emerged

as essential to applying the moment matching PMOR in the context of DC operating

point. The first issue is the computation of the orthogonal basis Q that will be used

in the projection. The second issue arises in the course projecting the nonlinear term

in (5.8) in the reduced space. The rest of this section takes a closer look into the

46

requirements needed in the implementation of the second issue, leaving the first issue

for Section 5.3.

Solving for the reduced system (5.6) for x(ξ) is typically done through applying the

Newton-Raphson iterative scheme. The core of this process requires the computation

of the reduced nonlinear function and its Jacobian matrix, which are, respectively,

obtained from,

f(x(ξ), ξ) = Q>f(Qx(ξ), ξ) (5.10)

and

J(x(ξ), ξ) =∂ f(x(ξ), ξ)

∂x= Q>J(Qx(ξ), ξ)Q (5.11)

Using the values obtained for x(ξ), it is possible to recover the values in x(ξ)

through the mapping Qx(ξ).

It is obvious from the procedure, as outlined above, that the main advantage of

using the proposed approach to track the variation of the DC operating point, is

having to factorize the reduced Jacobian matrix with size m, instead of the original

one which of size N and given that typically m << N .

However, the basic computation of the reduced nonlinear f ∈ Rm and J ∈ Rm×m

(as presented in (5.10) and (5.11), respectively) would still require invoking the full

circuit device model evaluation routines to compute the contribution to the N com-

ponents of f(·) as well as the nonlinear devices stamp on the N ×N original Jacobian

matrix J(·). This, of course, comes in addition to the O(mN) computational com-

plexity incurred in the linear algebra operations involving the projection Q. This

fact represents a significant disadvantage that could curb the computational savings

hoped for by entirely moving the computation from RN to the Rm domain.

The difficulty described above has already been elaborated upon in Section 4.4.3

and is a result of trying to apply the projection on f(x), the nonlinear term. The

47

idea of DEIM, briefly outlined in section 4.5, is meant to solve this difficulty. The

next section revisits this idea in the current context illustrating how it can be used

to alleviate the computational difficulty of projecting f(x).

5.2 Proposed MIP-Based Reduction

This section addresses the issue of alternating in computation between the original

domain RN and the reduced domain Rm with each point of ξ. The approach presented

in this regard is termed as the moment-based interpolation, or MIP, because it relies

on using an interpolatory matrix operator that is used by the DEIM framework.

However, it departs from the DEIM approach in the way in which the operator U is

constructed.

Similar to DEIM, MIP is premised on the idea that f (x(ξ), ξ) can be approximated

with another vector f (x(ξ), ξ) ∈ RN (same size), which is obtained as an interpolation

of k entries selected from the original f (x(ξ), ξ), with k << N . This notion is

formulated [19,105] more succinctly by defining

f (x(ξ), ξ) := Yfk (x(ξ), ξ) (5.12)

where,

• fk (x(ξ), ξ) is a set of k entries selected from, f (x(ξ), ξ) and best described using

a selector matrix P ∈ 0, 1N×k,

fk (x(ξ), ξ) = P>f (x(ξ), ξ) (5.13)

with P being a matrix whose k columns are defined as a set of k columns selected

48

from an N ×N identity matrix, i.e.,

P = [eγ1 , · · · , eγk ] (5.14)

with γi ∈ N, 1 ≤ γi ≤ N and eγi being the γthi column of an N × N identity

matrix.

• Y ∈ RN×k is defined in terms of P and another orthonormal matrix U

Y = U(P>U

)−1(5.15)

where it is assumed that P>U is invertible.

It should be obvious from (5.12) that the N entries of f(·, ·) are interpolations

of the k selected entries in f(·, ·). Furthermore, this interpolation becomes exact for

those k entries, in the sense that those entries in the former coincide with the same

entries in the latter, a fact that can be easily seen from,

P>f(x(ξ), ξ) = P>f(x(ξ), ξ) (5.16)

The interpolatory matrix operator in (5.12) then invites two issues to the fore-

front of the present development. The first issue is relevant to resolving problem of

alternating in computation between the reduced Rm and the full original model space

in RN . This issue is addressed by Section 5.2.1.

The second issue is concerned with constructing the selector matrix P and the

orthonormal basis U. The framework used for this purpose is predicated on the idea

that for any orthonormal matrix U ∈ RN×k, there is a selector matrix P ∈ 0, 1N×k

that can be constructed such that the error in∣∣∣∣∣∣f (x(ξ), ξ)− f (x(ξ), ξ)

∣∣∣∣∣∣ is minimized

[19]. The algorithm used to construct P given an arbitrary matrix U is considered

49

later in Section 5.3.2 based on [19].

On the other hand, the choice of U is typically made based on empirical grounds,

where it was observed that, for best results, the matrix U needs to span the subspace

spanned by f . In other areas of applications where the interpolatory operator has been

used, e.g. aerodynamics and fluid mechanics [20, 108, 109], U was obtained through

simulating the full nonlinear system that models the underlying problem domain and

using the results of the simulation to create a snapshot of the nonlinear part of the

model, which is analogous to f(x(ξ), ξ) in the present work. This process is repeated

many times to collect more snapshots that cover as much as possible the subspace

spanned by the nonlinear part. Upon termination, a singular value decomposition

(SVD) is run on the snapshots to extract the subspace that corresponds to the k

largest singular values and use that to construct the orthonormal basis U.

In the present application of tracking the DC operating point such a scenario is not

plausible for several reasons, foremost among them is the fact that number of snap-

shots required to cover the d-dimensional space spanned by the design parameters ξ

grows exponentially with the number of parameters d. This fact makes the snapshot-

based approach, even if it is possible to achieve, stand to erode the computational

gains anticipated in using the PMOR approach since it necessitates computing the

DC operating point at large number of points within the parameters space. In addi-

tion, the concept of snapshots defeats the very purpose of using a moment matching

approach to PMOR. This issue is addressed in Section 5.2.2 through proposing a new

method to construct U based on utilizing the concept of rooted trees.

5.2.1 Efficient Projection using MIP

This section illustrates the role that using f(x(ξ), ξ) instead of f(x(ξ), ξ) in (5.1)

plays in eliminating the problem of alternating in computation between the original

domain RN and the reduced domain Rm. To this end, and to make this illustration

50

lucid, the following assumptions will be temporarily made,

• f (x(ξ), ξ) has no explicit dependence on ξ. Therefore, henceforth, f (x(ξ), ξ)

will be replaced by f (x(ξ)).

• The N nonlinear functions in f (x(ξ)) are only component-wise functions of x.

Thus, f (x(ξ)) can be captured by,

f(x(ξ)) = [f1(x1(ξ)), · · · , fN(xN(ξ))]> . (5.17)

Next, replace f(x(ξ)) with f(x(ξ)) in (5.1), and x with Qx, pre-multiplying both

sides by Q> to obtain the new expression for f (x(ξ)) (different than the one in (5.10))

that is given by

f(x(ξ)) = Q> f(x(ξ))

= Ψ Ξ(x(ξ)) (5.18)

where Ψ ∈ Rm×k is a constant matrix given by

Ψ = Q>Y (5.19)

and Ξ(x(ξ)) ∈ Rk is given by

Ξ(x(ξ)) = fk(Qkx(ξ)) (5.20)

with fk ∈ Rk and Qk ∈ Rk×m being, respectively, the k elements of f(·) and k rows of

Q selected by P>.

It is obvious from (5.18) that computing the reduced nonlinear function f(x(ξ)) of

(5.18) can be carried out entirely in a reduced-domain Rk, since Ψ is a constant m×k

51

matrix, and evaluation of Ξ(x(ξ)) only requires using a constant matrix Qk ∈ Rk×m

obtained by selecting k rows from the matrix Q and evaluation of k components of

the nonlinear function. Thus, utilizing the selector matrix P of (5.14) in (5.18) is the

key idea that enables one to avoid computations proportional to O(N).

Similar arguments can be made to show that computing the reduced Jacobian

matrix (5.11) can also be entirely performed in a reduced domain using (5.18) as

evidenced by,

J (x(ξ)) =∂ f(x((ξ)))

∂x= Ψ︸︷︷︸

m×k

∂fk∂xk︸︷︷︸k×k

Qk︸︷︷︸k×m

, (5.21)

where xk = P>x is the k components of x selected by P>. Therefore, computing the

reduced Jacobian can be executed while avoiding the type of O(N) computational

complexity that arises from direct projection.

5.2.2 Orthonormal Basis U Based on Moment Matching

Computing U in the present work is based on utilizing the concept of the rooted trees

to compute the moments of f(x(ξ)). To simplify the mathematical illustration of the

basic ideas involved in this process, we will consider, without any loss of generality,

the case of a single design parameter ξ (d = 1). Proceeding forward, we posit that

f(x(ξ)) varies smoothly enough in a neighbourhood around the nominal design point

ξ0 to allow a Taylor series expansion of f(x(ξ)) at ξ = ξ0,

f(x(ξ)) =∞∑

i=0

Fi(ξ0) (ξ − ξ0)i (5.22)

where, Fi(ξ0) are referred to as the moments of f(x(ξ)) and are given by

Fi(ξ0) =1

i!

dif(x(ξ))

dξi

∣∣∣∣ξ=ξ0

(5.23)

52

The matrix U is then taken to span the column span of the first k moments of

f(x(ξ)), i.e.,

U = colspan

[F0(ξ0) F1(ξ0) · · · Fk−1(ξ0)

](5.24)

In computing Fi(ξ0), one should note that since f(x(ξ)) depend on x(ξ), then its

moments (F1(ξ0)) will implicitly be determined by the moments of x(ξ). The moments

of x(ξ) were introduced earlier in Section 5.1 in the general Taylor expansion of x(ξ)

of (5.2), which for the case of a single design parameter used here for illustration

simplifies to,

x(ξ) =∞∑

i=0

Mi(ξ0) (ξ − ξ0)i (5.25)

The implicit dependence of Fi(ξ0) on Mi(ξ0) suggests that in order to compute

Fi(ξ0), Mi(ξ0) need to be computed first. On the other hand, (5.3) shows that Mi(ξ0)

are also needed to compute Q. Therefore, both matrices U and Q are obtained by

first computing the moments Mi(ξ0). The process of computing Mi(ξ0) is based on

the notion of rooted tree and will be detailed in the next section.

5.3 Computing the Moments Using Rooted Trees

The objective in this section is to compute the moments of x(ξ), or Mi(ξ0) and use

that to construct the matrices U used for the iterpolatory operator and Q used in

the projection operations described in Section 5.2.1. Computing the matrix U then

enables computing the matrix P, thereby completing the projection process.

53

5.3.1 Computing the Moments Mi(ξ0) and Fi(ξ0)

Computing the moments Mi(ξ0) is carried out first through differentiation of (5.1)

with respect to ξ, and the substitution of x(ξ) from (5.25). This process yields,

(G(ξ) + J(x(ξ))

)∑

i=1

iMi(ξ0)(ξ − ξ0)i−1 +dG(ξ)

dξ

∑

i=0

Mi(ξ0)(ξ − ξ0)i = 0 (5.26)

where J(x(ξ)) = ∂f(x(ξ))/∂x is the Jacobian matrix of the full circuit model.

Expanding J(x(ξ)) and G(ξ) as a Taylor series around ξ = ξ0,

J(x(ξ)) =∑

i=0

Ji(ξ0) (ξ − ξ0)i (5.27)

G(ξ) =∑

i=0

Gi(ξ0) (ξ − ξ0)i (5.28)

with Gk(ξ0) and Jk(ξ0) being the kth (matrix-valued) Taylor series coefficient of G(ξ)

and J(x(ξ)), respectively, when expanded around ξ = ξ0. Define

Jk(ξ0) = Gk(ξ0) + Jk(ξ0) (5.29)

In the following, we shall use Jk,Gk and Mk in place of Jk(ξ0),Gk(ξ0) and Mk(ξ0),

respectively, to make the mathematical analysis terser.

Substitute from (5.27), (5.28) and (5.29) in (5.26)

∑

i=1

∑

j=0

iJjMi (ξ − ξ0)i+j−1 = −∑

i=0

∑

j=1

jGjMi (ξ − ξ0)i+j−1 (5.30)

If we differentiate both sides with respect to ξ for p times, we get

∑

i=1

∑

j=0

i(i+ j − 1)(i+ j − 2) · · · (i+ j − p)JjMi (ξ − ξ0)i+j−1−p

54

= −∑

i=0

∑

j=1

j(i+ j − 1)(i+ j − 2) · · · (i+ j − p)GjMi (ξ − ξ0)i+j−1−p (5.31)

The above equation is valid for all ξ. Thus setting ξ = ξ0, all terms for which

i+ j − 1− p > 0 will vanish leaving only the terms for which

i+ j − 1− p = 0 (5.32)

On the left side of (5.31), this will lead to indices given by

i = 1, 2, · · · , p+ 1

j = p, p− 1, · · · , 0 (5.33)

On the right-side of (5.31), the indices i and j are given by

i = 0, 1, · · · , p

j = p+ 1, p, · · · , 1 (5.34)

The above two relations show that on both the left- and right-side, i and j are related

by j = p− i+ 1, which upon substitution into (5.31) yields

p+1∑

i=1

i (p)(p− 1) · · · (1)︸ ︷︷ ︸p!

Jp−i+1Mi =

−p∑

i=0

(p− i+ 1) (p)(p− 1) · · · (1)︸ ︷︷ ︸p!

Gp−i+1Mi (5.35)

Rearranging, we get

(p+ 1)J0Mp+1 +

p∑

i=1

iJp−i+1Mi =

55

− (p+ 1)Gp+1M0 −p∑

i=1

(p− i+ 1)Gp−i+1Mi (5.36)

Therefore,

J0Mp+1 = −p∑

i=1

(i

p+ 1Jp−i+1 +

(p− i+ 1)

p+ 1Gp−i+1

)Mi −Gp+1M0 (5.37)

It is obvious that (5.37) provides a recursive formula to compute Mp+1(ξ0) from

Mk(ξ0), k ≤ p. It should be also clear that the main computational efforts in com-

puting Mi(ξ0) is a result of having to run the LU factorization of the Jacobian matrix

J0 and use several forward/backward substitutions with different right-side vectors.

However, given that the matrix J0 is the same matrix used in solving for the DC op-

erating point at the nominal values for the design parameters, ξ, its LU factors will

be readily available upon convergence to the nominal DC operating point. Hence,

the only extra computational cost required to compute Mi(ξ0) is mainly the for-

ward/backward substitutions, in addition to computing the Taylor coefficient matri-

ces Jm, which is discussed next.

Clearly J(x(ξ)) is a function of x, and therefore Ji(ξ0) is also a function of Mi(ξ0).

In fact, it can be shown that Jk(ξ0) is a function of all Mp(ξ0), for p ≤ k [110].

Handling the process of computing Ji(ξ0) based on Mi(ξ0) has been introduced in

the literature and carried out through the notion of rooted trees [55,110]. This idea is

used to represent each nonlinear function entry in the matrix J(x(ξ)) as a tree, with

the root node of the tree representing the expression of the nonlinearity itself and

the leaf nodes representing the values in Mi(ξ0) or the constants in the expression

of the nonlinear function. Other intermediate nodes in the tree represent atomic

nonlinear functions such as the exponential function or arithmetic operators such

multiplication. The process of computing the value of an entry in Ji(ξ0) is done by

traversing the rooted tree in the upward direction starting with the root node, with

56

each node simply triggering the node the next level. When a leaf node is reached,

it sends back to the nodes at the lower level its assigned value, which could be a

value in the vector Mi(ξ0), computed earlier, or the value of the constant assigned to

this leaf node. The tree is next traversed in the downward direction with each node

computing a pre-specified recursive expression with inputs taken from the nodes at

the upper level in the tree. The process is continued downwards, and upon arriving

back at the root node, yields the required value in Ji(ξ0).

The above process for computing Mi(ξ0) can be used to compute Fi(ξ0). Similar

to Jk(ξ0), Fk(ξ0) is a function of all Mp(ξ0) for all p ≤ k. This makes Fi(ξ0) function

of Mi(ξ0). In a similar manner, rooted trees can be used to compute Fi(ξ0) by

representing the nonlinear expressions entries in f(x) as the root node in a tree with

the leaf nodes representing the Mi(ξ0) or other constants that might be used by the

nonlinear function components in f(·).The process is first started with the computation of M0(ξ0), which, as can be seen

from (5.2), is the DC operating point at the nominal values of the design parameters.

M0(ξ0) is then assigned to the leaf nodes on the rooted trees representing the expres-

sions of both J(x) and f(x). The trees are traversed in both directions as mentioned

above to obtain F1(ξ0) and J1(ξ0), which are then used to compute M1(ξ0) using the

recursive formula in (5.37). Again, both of M0(ξ0) and M1(ξ0) are assigned to the

leaf nodes, and traversing the tree is repeated again resulting in F2(ξ0) and J2(ξ0),

and so forth. Further details on the construction and operation of the rooted tree can

be found in [55,110,111].

Rooted Trees

For the illustration of the idea of rooted trees, consider the circuit at the left bottom

corner of Figure 5.1. In the MNA formulation, this circuit has N = 1, with the diode

current assumed to be related to the node voltage through, i(v(t)) = I0

(ev(t)/VT − 1

).

57

Let ξ represent the thermal voltage VT . Thus, it should be straightforward to write

for this circuit

JI (zp(ξ), ξ) =I0

ξe

zp(ξ)

ξ (5.38)

To explain the operation of rooted trees, we assume that M0 and M1 have been

computed and the goal now is to compute M2 using (5.37) where J1 will be needed

on the right side to accomplish this goal. The rooted tree representation of (5.38)

will be used for that purpose. The process is started by first computing J1. The

rooted tree representation of JI (·, ·) is the technique that is used to compute J1. The

structure to the right of the circuit in Figure 5.1 is the rooted tree corresponding to

the expression in (5.38).

x

÷ e

÷

1

23

456

78ww

C w

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<latexit sha1_base64="4JG1BLGcgaMaO3JksJYYwjRtcB0=">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</latexit>

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58

Figure5.1:AnexamplecircuitanditsrootedtreerepresentationforJI(x(ξ),ξ).

InFigure5.1,thenodesinthegraphdefineanarithmeticoperations. Thus,

node1correspondstoamultiplication,2anexponentialfunction,3and4adivision,

etc.EachnodehasalocalstoragethatcontainsthevaluesoftheTaylorcoefficients

correspondingtothetermrepresentedbythenode.Forexample,node2storesthe

Taylorcoefficientsofezp(ξ)/ξinξ.Leafnodeswith intheirtableentriesshowwhich

Taylorcoefficientsareknown,assumingM0andM1arecomputedalready.Ascan

beseeninthetreestructureinFigure5.1,nodesinthetreecanbeclassifiedas,root

(parent)nodethatrepresentstheexpressionin(5.38),leafnodes(triangles)whichare

59

the terminal (childless) nodes, and intermediate nodes (circles). Leaf nodes represent

either the constant terms in the expression, e.g., I0, or the terms that have explicit1

or implicit (indirect)2 dependence on the parameter ξ, which are marked with “w”.

Intermediate nodes represent atomic functions, such as the exponential function, or

arithmetic operators, e.g. division or multiplication.

The fundamental ideas of operation in the rooted trees can be summarized as

follows. With exception of the leaf nodes, all nodes in the tree compute the Taylor

coefficients of the part of the expression they represent, based on the Taylor coeffi-

cients of their children nodes and their own lower-order Taylor coefficients. Nodes

with constant terms or explicit dependence on ξ have known Taylor coefficients in ξ,

e.g., nodes 6, 5, and 7 in Figure 5.1. Each node maintains a local storage (shown as

a table in the figure) which it uses to store its own Taylor coefficients. The process

starts by first assigning the Taylor coefficients to the leaf nodes, as mentioned. Com-

puting J1 is triggered at the root node, which instigates its immediate descendant, so

to speak, to compute its 1-st order Taylor coefficient. The tree is traversed in the up-

ward direction, with each node simply triggering the node at the next level prompting

it to compute the 1-st coefficient. When a leaf node is reached, it responds by sending

back to the nodes at lower level the values already stored in the local storage table.

The tree is then traversed in the downward direction, with each node computing, us-

ing a pre-specified expression (specific to each node type), the value of the 1st Taylor

coefficient at that node. The computed value is stored locally to the node (for future

use), and sent down to the lower level. The process is continued downwards, and

upon arriving back at the root node, will have accumulated in its path the required

value of J1. Therefore, M2 can now be computed and the process is repeated to

compute J2, and so forth. Further details on the automated construction, operation

1such as nodes 5 and 7 representing ξ.2e.g., node 8, representing x(ξ).

60

of the rooted tree can be found in [55, 110, 111]. Once the moments up to order m

have been computed a “skinny” QR factorization [107] can be performed on them to

obtain V.

5.3.2 Construction of the Matrix P

Once U is obtained using the rooted trees as explained in the previous subsection,

then P can be constructed through U using the algorithm proposed in [19] and shown

in the pseudo-code representation of Algorithm 3 for completeness. The operator

max employed in this algorithm is the same as the Matlab function max(u), which

returns the maximum element in a vector and its location (index). The output of the

algorithm is a set of k indices γi,∈ N, with i = 1, · · · , k that represents the set of

columns selected from the identity matrix use to build P. One needs to stress here

that the resulting P from the above algorithm is U-dependent.

Algorithm 3: Indices selection algorithm

input : ulkl=1 linearly independentoutput: Γ = γi, i = 1, · · · , k

1 [ρ1, γ1] = max (u1);2 Γ← γ1;3 for l← 2 to k do4 Solve

(P>U

)c = P>ul for c;

5 r = ul −Uc;6 [ρl, γl] = max (r);7 U← [U ul] , P← [P eγl ], Γ← Γ ∪ γl;8 end

5.4 Extension to General Nonlinearity

The previous development used a special form of nonlinearity in which each compo-

nent in f(·) was assumed to be a function of only one independent variable in x. The

61

objective was to facilitate presenting the core ideas without having to utilize cumber-

some notations. Handling the more general nonlinear f(x(ξ), ξ) is easily implemented

by noting that in a typical nonlinear circuit with many variables the elements in f in-

dexed by the set Γ, i.e. fγi(x(ξ)) are typically dependent on a few number of variables

in x. This fact can be captured by the introducing the following structures.

• A set jγi of integer values with pγi members (pγi = |jγi|) that is comprised from

the indices of the variables that control the expression of fγi(x(ξ)),

• A local selector matrix Lγi ∈ 0, 1N×pγi . Columns in Lγi has “1” in the row

corresponding to the elements in jγi , and “0” otherwise, and

• A global selector matrix L ∈ 0, 1N×p, where

p = Σki=1 |jγi| ,

given by

L =

[Lγ1 Lγ2 · · · Lγk

](5.39)

With the above notation in place, (5.12) can be extended to general nonlinearity

using

f(x(ξ)) = U(P>U

)−1P>f(L>Qx(ξ)) (5.40)

Proceeding with manipulation similar to what was followed in Section 5.2.1, the

reduced nonlinear vector follows from,

f(x(ξ)) = Ψ Ξ(x(ξ)) (5.41)

where Ξ(x(ξ)) in this case is given by

Ξ(x(ξ)) = fk(Qpx(ξ)) (5.42)

62

with Qp being the p rows of Q selected by L>. Similarly, the reduced Jacobian matrix

is given by the (more general) formula

J (x(ξ)) = Ψ︸︷︷︸m×k

∂fk∂xp︸︷︷︸k×p

Qp︸︷︷︸p×m

, (5.43)

Chapter 6

Numerical Examples

Three examples are presented in this section. The first example presents a proof of

concept and demonstrates the accuracy of the proposed approach compared with the

direct sensitivity approaches.

The other two examples validate the accuracy and efficiency of the proposed ap-

proach. An error measurement based on Equation (6.1) was adopted to illustrate the

accuracy of the proposed algorithm.

ε =||xfull

DC − xproposedDC ||2

||xfullDC||2

(6.1)

xfullDC denotes the original circuit’s DC operating point, and xproposed

DC is the DC

operating point evaluated using the proposed procedure described in this chapter.

A selected set of d parameters in each of the two circuits are marked as the

design parameters represented by the parameters ξ in the formulation. Each of the

parameters are assumed to vary between -20% to +20% around their nominal design

values, generating 2d “corners” in the Rd design parameter space. The DC operating

point, evaluated using the original and reduced systems, is obtained, and the error

between the two solutions at each parameter space corner is evaluated.

To implement the proposed method, k and m are defined for each example. k is

63

64

the number of columns of the orthonormal matrix U and selector matrix P. In other

words, it defines the number of entries chosen in f(x(ξ), ξ) for MIP. m defines the

order of the reduces system after applying PMOR.

65

6.1 Example 1: CMOS Operational Amplifier

The first example is the MOSFET-based OpAmp circuit shown in Figure 6.1. The

variation in the DC operating point in this circuit is the result of variations in the

length and width of the channel in several transistors. The nominal values of the

length (L) and width (W ) are provided in the circuit schematic (indicated by the

ratio W/L in µm). For this example VDD=2.5V.

Figure 6.1: Schematic of operational amplifier.

Figure 6.2 shows the change in the DC operating point of the output node versus

variations in the channel lengths of all the transistors. Likewise, Figure 6.3 shows

the change in the DC voltage of the same node versus variations in the width of

transistors M1, M2, M5, M7, M8. The variations illustrated in both figures also

show a comparison for results obtained through different methods: the first approach

66

(dubbed “original”) refers to the exact variations which are computed by solving the

original system of nonlinear equations for the DC operating point; the second method

is the proposed approach obtained with a reduced system with m = 6 moments;

the third approach (labeled as “direct Sensitivity”) is based on using the computed

moments Mi(ξ0), i = 1, 2, · · · , 6 to evaluate the variations in DC solution.

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6Channel length ( m)

-1

0

1

2

3

4

5

6

DC

Vol

tage

OriginalProposedDirect Sensitivity

Figure 6.2: Variations of the DC voltage at the output node of OpAmp vs. variationsin the length of the MOSFETs channels.

67

80 100 120 140 160 180Channel Width ( m)

-0.9

-0.85

-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

-0.5

-0.45

DC

Vol

tage

OriginalProposedDirect Sensitivity

Figure 6.3: Variations of the DC voltage at the output node of OpAmp vs. variationsin the width of the MOSFETs channels.

The above results indicate that the reduced system, created through the proposed

approach, can accurately capture the variations in the DC operating point with up

to 200% variations in the design parameters around their nominal values.

68

6.2 Example 2: 741 Op-Amp

The circuit used in this example is an inverting amplifier based on the µA741 OpAmp,

whose schematic is shown in Figure 6.4. The internal schematic of the µA741 is

presented in Figure 6.5. As seen from Figure 6.5, the circuit’s DC bias is provided by

two power supplies VCC=15V and VEE=−15V.

47k+

_

741

27k 1.2k

Figure 6.4: Schematic of inverting amplifier

VEE

VCC

In(+) In(-)Vout

Figure 6.5: Internal schematic of µA741 OpAmp [1].

69

The resistors circled in Figure 6.5 are considered as the design parameters ξ1 and

ξ2. Each parameter is varied by ±20%. Table 6.1 provides the error computed using

(6.1) at the four possible corners in the parameter space. In this example, k was

defined as 5, whereas the size of the reduced system using PMOR is m = 6.

Table 6.1: Error at each corner case for the DC simulation of the inverting amplifier.

Param Combination (ξ) ε in %

(0%, 0%) 1.125× 10−11

(20%, 20%,) 2.58× 10−2

(−20%, 20%,) 1.36× 10−2

(−20%,−20%) 2.16× 10−2

(−20%,−20%) 1.21× 10−2

Table 6.2 demonstrated the reduction in the size of the Jacobian matrix (J) due

to PMOR. By setting m as 6, the size of the system was reduced from 202 to 6.

The advantages of MIP do not only stem from the reduction in the size of the

Jacobian matrix used during factorization. It is also a result of drastically reducing

the number of nonlinear model evaluations, made possible through the projection

operator P. Table 6.3 demonstrates the optimization achieved by only evaluating

selected nonlinear functions using MIP.

Table 6.2: A comparison between the size of the original system and the reducedmodel using PMOR.

Original systemSize of X(ξ) 202x1

Size of J(ξ) 202x202

Reduced System(m = 6)

Size of X(ξ) 6x1

Size of J(ξ) 6x6

70

Table 6.3: A comparison of number nonlinear function evaluation between the originalsystem and DEIM.

Original systemSize of f(x(t, ξ), ξ) 202x1

NL function evals in f(x(t, ξ), ξ) 75

Reduced System(k = 5)

Size of fk(x(t, ξ), ξ) 5x1

NL function evals in f(x(t, ξ), ξ) 5

The impact in the reduction of nonlinear function evaluations are further high-

lighted when considering the total number of N-R iteration used to solve the DC

solution and all the parameter combinations that must be evaluated. Table 6.4

demonstrates this by showing the total number of nonlinear device model evalua-

tions in both the original and reduced models, indicating a significant reduction in

the number of times the device model had to be evaluated. The “Total Evaluations”

was calculated by adding the number of nonlinear function evaluations in f(x) and

J(x) and multiplying it by the total number of N-R iterations required to evaluate

all parameter combinations.

Table 6.4: A comparison between the savings achieved during nonlinear functionevaluation using traditional method vs. the proposed method.

# of NL function eval. TotalIter.

TotalEvals.

f(x(t, ξ), ξ) ∂f(x(t,ξ),ξ)∂x(t,ξ),ξ)

Orig. 75 175 36 9,000

Prop. 5 12 25 425

71

6.3 Example 3: Power Distribution Network

(PDN)

The circuit in this example is based on a Power Distribution Network (PDN)

conceptually represented by Figure 6.6. A PDN is a grid used on Silicon to distribute

power to the various devices in an integrated circuit. The PDN is typically modelled

by a mesh of resistors with capacitors connected to Ground. The circuit in this

example uses a PDN, modelled by a mesh of resistors with 10,494 nodes, where the

nodes are connected to the DC supply rail of inverter circuits via resistors. There

is a total of 10 SN7404 inverters (Figure 6.7) used in this circuit. The inverters are

connected to the ground plane via resistors.

Figure 6.6: Power distribution network [2].

72

Figure 6.7: Circuit schematic of the SN7404 inverter [3].

Four design parameters were used in this example: two parameters were used to

represent the group of resistors connecting the PDN nodes to the supply rail of the

inverters and the group of resistors connecting the inverter ground node to the ground

plane. The other two parameters are taken from the two resistors circled in Figure

6.7 in all the inverters. This configuration resulted in 16 corners in the design space.

In this example, k was chosen to be 5, and m was set to 7. The accuracy of

the reduced system in approximating the variation in the DC operating point of the

original system is illustrated by Table 6.5. It tabulates the error in the DC operating

point obtained at the 16 corners in the design space.

73

Table 6.5: Error at each corner case for the DC simulation of the PDN.

Param Combination (ξ) ε in %

(0%, 0%, 0%, 0%) 5.13× 10−11

(+20%,+20%,+20%,+20%) 7.97× 10−6

(−20%,+20%,+20%,+20%) 6.62× 10−6

(−20%,−20%,+20%,+20%) 1.55× 10−6

(−20%,−20%,−20%,+20%) 1.55× 10−6

(−20%,−20%,−20%,−20%) 1.55× 10−6

(+20%,−20%,−20%,−20%) 3.49× 10−7

(+20%,+20%,−20%,−20%) 3.78× 10−6

(+20%,+20%,+20%,−20%) 7.97× 10−6

(+20%,−20%,−20%,+20%) 3.49× 10−7

(−20%,+20%,+20%,−20%) 6.62× 10−6

(+20%,−20%,+20%,+20%) 2.45× 10−6

(+20%,+20%,−20%,+20%) 3.77× 10−6

(−20%,+20%,−20%,+20%) 3.07× 10−6

(+20%,−20%,+20%,−20%) 2.45× 10−6

(−20%,−20%,+20%,−20%) 1.55× 10−6

(−20%,+20%,−20%,−20%) 3.07× 10−6

Table 6.6 demonstrates the reduction in the size of the Jacobian matrix (J) due

to PMOR. By setting m as 7, the size of the system was reduced from 10706 to 7.

As explained previously, MIP also drastically reduced the number of nonlinear

model evaluations, which is made possible through the projection operator P. Table

6.7 presents the optimization observed by only evaluating selected nonlinear functions

using MIP.

74

Table 6.6: A comparison between the size of the original system and the reducedmodel using PMOR.

Original systemSize of X(ξ) 10706x1

Size of J(ξ) 10706x10706

Reduced System(m = 7)

Size of X(ξ) 7x1

Size of J(ξ) 7x7

Table 6.7: A comparison of number NL function evaluation between the originalsystem and DEIM.

Original systemSize of f(x(t, ξ), ξ) 10706x1

NL function evals in f(x(t, ξ), ξ) 150

Reduced System(k = 5)

Size of fk(x(t, ξ), ξ) 5x1

NL function evals in f(x(t, ξ), ξ) 5

The impact in the reduction of nonlinear function evaluations are further high-

lighted when considering the total number of N-R iteration used to solve the DC so-

lution and all the parameter combinations that must be evaluated. Table 6.8 demon-

strates this by recording the total number of nonlinear device model evaluations in

both the original and reduced models. It demonstrates a significant reduction in the

number of times the device model had to be evaluated.

Table 6.8: A comparison between the savings achieved during NL function evaluationusing traditional method vs. the proposed method.

# of NL function eval. TotalIter.

TotalEvals.

f(x(t, ξ), ξ) ∂f(x(t,ξ),ξ)∂x(t,ξ),ξ)

Orig. 150 330 89 42,720

Prop. 5 13 95 1,710

75

6.3.1 Summary

As demonstrated using the numerical examples in this Chapter (6), the proposed

method is able to accurately and efficiently simulate nonlinear circuits. For both

examples it was demonstrated that the advantages in the MIP approach do not stem

merely from reducing the size of the Jacobian matrix that needs to be factorized.

It also results from drastically reducing the number of nonlinear model evaluations

which is made possible through the projection operator P.

Chapter 7

Conclusion

7.1 Concluding Remarks

The main objective in this thesis has been to wield the idea of Model Order Reduction

(MOR) to tackle the problem of repeated DC analysis of nonlinear circuits.

The problem of DC analysis considered in the thesis refers to the task of repeating

the computation of the DC quiescent point at values for a selected set of design

parameters, different than their nominal values.

The approach developed in the thesis is based on the ideas of Discrete Empir-

ical Interpolation Method (DEIM) and concepts of moment-matching model-order

reduction techniques.

The unique contribution of the thesis lies in adapting DEIM and moment matching

to the context of the DC problems. More particularly, the notion of rooted trees was

developed to enable the construction of the projection matrices used in the reduction

process.

Through the concept of rooted trees, it was demonstrated that it is possible to con-

struct the projection basis at the nominal design values and without having to create

the so-called “snapshots” that necessitated simulating the full system at numerous

points in the parameter space.

76

77

Several examples have been presented to demonstrate the validity, accuracy and

efficiency of the proposed methodology.

7.2 Future Work

The ideas and approaches developed in this thesis are sufficiently broad to be used

in tackling different problems and in different domains of applications. Indeed, the

DC nonlinear equations, which represented the core and target of this thesis had

no significant restriction in their formulation that would limit the applicability of

the proposed approach to the DC problem. This observation represents the main

motivation in pursuing problems that posit similar questions to the DC problem and

the parameter variations.

One of the main problems that will be the immediate focus in future research is

the problem of Harmonic Balance (HB). In HB, as in the DC problems, the goal is to

solve a set of nonlinear equations but with a more complex structure. It also happens

that in HB problems there are parameters that need to varied from their nominal

design values and the HB analysis needs to be repeated at those new values.

The ideas presented in this thesis are ideal tools to tackle the HB problem. In

fact, the author has developed initial prototypes that showed significant efficiency for

the HB problem and plans to pursue those ideas in his Ph.D. program. A sample of

the results obtained is described in the following example to demonstrate the validity

of using the methodology in this thesis for HB problems.

7.2.1 Example: Low Noise Amplifier

The Low Noise Amplifier (LNA) with the schematic shown in Figure 7.1 is considered

for this example, where the Eber-Moll model has been used to model the bipolar

junction transistors. The steady-state output voltage in response to a sinusoidal

78

input of 0.2V amplitude is shown Figure 7.2.

Figure 7.1: LNA schematic.

Figure 7.2: Steady-state of the output response.

79

Two parameters ξ1 and ξ2 are used in this example: ξ1 controls the values of circled

capacitors in Figure 7.1 using the relationship in Equation (7.1) and ξ2 controls the

value of the circled inductor L1 using the relationship in Equation (7.2).

C(ξ1) =ε(L(1 + ξ1)×W (1 + ξ1))

d(7.1)

L1(ξ2) = L(1 + ξ2) (7.2)

The Proposed HB prototype based on PMOR was used in this example to perform

the parameter sweep on the first, second, and third harmonics of the steady-state

output voltage. This produced the results depicted in the bottom of the plots of

Figures 7.3-7.5 with a CPU time of 1.63 mili-seconds. Figures 7.3-7.5 also demonstrate

the accuracy of the results by comparing them with the results obtained from the full

HB parameter sweep which required 0.185 seconds. This indicates that the proposed

approach is around 113 times faster than the conventional parameter sweep of the

full HB.

80

Figure 7.3: Comparison between the proposed method and traditional HB for thevariation in the first harmonic of the output node vs. the variation in the designparameters

Figure 7.4: Comparison between the proposed method and traditional HB for thevariation in the second harmonic of the output node vs. the variation in the designparameters.

81

Figure 7.5: Comparison between the proposed method and traditional HB for thevariation in the third harmonic of the output node vs. the variation in the designparameters.

List of References

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Univ. Press, seventh ed., 2014.

[2] S. R. Nassif, “Power grid analysis benchmarks,” in 2008 Asia and South Pacific

Design Automation Conference, pp. 376–381, 2008.

[3] Texas Instruments Inc., Dallas, TX, USA, Hex Inverters Datasheet, 1983.

[4] R. C. Melville, L. Trajkovic, S.-C. Fang, and L. T. Watson, “Artificial param-

eter Homotopy methods for the DC operating point problem,” IEEE Trans.

Comput.-Aided Design Integr. Circuits Syst., vol. 12, pp. 861–877, June 1993.

[5] M. Striebel and J. Rommes, “Model order reduction of nonlinear systems in

circuit simulation: Status and applications,” in Model Reduction for Circuit

Simulation (P. Benner, M. Hinze, and E. J. W. ter Maten, eds.), vol. 74 of

Lecture Notes in Electrical Engineering, ch. 17, pp. 289–301, Berlin, Heidelberg,

Germany: Springer-Verlag, 2011.

[6] D. De Jonghe and G. Gielen, “Characterization of analog circuits using transfer

function trajectories,” IEEE Trans. Circuits Syst. I, vol. 59, pp. 1796–1804,

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Appendix A

Fundamental Notions

A few fundamental notions from linear algebra related to the subject of this thesis

that are repeatedly used are briefly formalized. For more details any fundamental

linear algebra references, e.g. [112–116] to name a few, can be consulted with.

Definition A.1. subspace: A subspace of a vector space Rn is a subset V of Rn

that has three following properties and hence, is is a vector space in its own right.

• It includes the zero vector; it is 0 ∈ V,

• ∀ vi,vj ∈ V, we have vi + vj ∈ V,

• ∀ vi ∈ V, α ∈ R , we have αvi ∈ V,

Definition A.2. span: Given a set of vectors v1,v2, . . . ,vm, the set of all

linear combinations of these vectors is a subspace referred to as the ”span” of

v1,v2, . . . ,vm. It is

span v1,v2, . . . ,vm =

v∣∣∣ v =

m∑

i=1

βivi ∀βi ∈ R

⊂ Rn (A.1)

Definition A.3. Column Spaces of a Matrix: Let V = [v1,v2, . . . ,vm] be an

n × m matrix. The colsp of matrix V is the set of all linear combinations of the

94

95

columns of matrix V. Hence, it is

colsp(V) = span v1,v2, . . . ,vm (A.2)

Definition A.4. Basis: Given a subspace defined as span v1,v2, . . . ,vm ⊂ Rn,

the set of vectors v1,v2, . . . ,vm are a basis for the subspace if they are a linearly

independent set.